Your question is multi-faceted. Starting with the premise that the type of vehicle that Virgin Galactic currently operates is capable of point-to-point suborbital flight. This is probably not a correct assumption, as I will explain later, although avoiding too much detail since this is Aviation not Space Exploration. But the more important part of your question involves the clearances required to land a spaceplane at a commercial airport. In fact there are already some FAA approvals in place for this. In May 2022 the FAA gave approval for Huntsville International Airport in Alabama to be used as a landing site for the Sierra Space Dream Chaser orbital spaceplane. This is separate from the license that Sierra Space will need to obtain from the FAA for these landings, which itself is separate from the Vulcan Centaur launch license. The procedures and clearances for suborbital point-to-point landings at a commercial airport would presumably be similar to those needed for the Dream Chaser landings, so we'll take a look at what those are.
The current plan, or at least the proposal by Sierra Space, is that Dream Chaser will land at Huntsville when returning from ISS (International Space Station) cargo missions. There will be no astronauts on these flights. An Environmental Assessment has been completed, and the FAA is currently working with Sierra Space on their proposal to conduct one landing per year at Huntsville in 2023, 2024, and 2025, two landings in 2026, and three landings in 2027. Although a landing in 2023 is no longer possible, the timeline gives an indication that all parties involved expect that the flight rate for landings at a commercial airport can start out slow then ramp up as experience is gained. Also the number of Dream Chaser flights to ISS will be limited, so I wouldn't take these flight rates to indicate the maximum number of flights that a point-to-point vehicle would be allowed to fly.
Dream Chaser's first flight to space will be a combination test flight and cargo flight to ISS, currently scheduled for 1st quarter 2024. The first flight will presumably land at the Kennedy Space Center in Florida as previously planned. It's not clear when the first Dream Chaser landing at Huntsville would take place. I suspect that decision might not be made until after a successful completion of the first flight.
The FAA documents don’t go into detail about airspace closures, etc, as that will apparently be dealt with in the actual launch licenses.
Under the Proposed Action, the FAA would also issue Letter(s) of Agreement to HSV and Sierra Space to outline notification procedures prior to, during, and after an operation and procedures for issuing a Notice to Airmen.
I would expect there would be some similarity to the arrival of Air Force One, with brief closures of the airport and surrounding airspace. Although the big difference here would be that due to its steep descent Dream Chaser would pass very quickly through Class A airspace (18,000 to 60,000 feet), and it would spend very little time in the Class C airspace surrounding Huntsville International. As both ATC and Sierra Space gain experience with these landings, hopefully disruptions to other traffic would be minimized.
The environmental review goes into quite a bit of detail about the procedures involved, so I won't go into all of the details here other than these highlights:
The Dream Chaser vehicle would remain above controlled airspace for the majority of the overflight of
New Mexico, Texas, Oklahoma, Arkansas, Louisiana, Mississippi, Florida, and Alabama. The reentry vehicle
would descend below 60,000 feet altitude above mean sea level (MSL) approximately 10-20 miles from
HSV prior to landing (15 nm southwest of HSV) and would be operating below 60,000 MSL for about
three to four minutes. The Dream Chaser would slow to subsonic speed at an altitude of approximately 35,000 feet while still flying northeast. The vehicle would turn directly onto an extended straight in final approach portion that commences approximately 8.5 nm from Runway 18L-36R threshold at an altitude of approximately 19,000 feet above MSL.
Dream Chaser landings would take place on the 10,000 foot runway 18L/36R at Huntsville.
Huntsville International Airport and proposed reentry site (Source: FAA)
What is interesting is that this is not the longest runway at Huntsville, 18R/36L is longer at 12,600 feet. Apparently 10,000 feet is long enough for Dream Chaser, and presumably using this runway will be the least disruptive to other aircraft, since this runway will be closed from two hours before the deorbit burn until Dream Chaser is removed from the runway eight hours after landing, a total period of about eleven hours. Even though Dream Chaser uses non-toxic RCS fuel and can be approached by ground crew immediately after landing, it is currently planned to allow eight hours for propellant safing before Dream Chaser can be towed from the runway. I am guessing that this is a very conservative estimate, and that after a few landings the amount of time that Dream Chaser is sitting on the runway after landing will be quite a bit less.
Meanwhile runway 18R/36L will remain in operation for other aircraft during this time, except for a one hour period surrounding the landing, beginning 15 minutes before the deorbit burn until Dream Chaser lands about 45 minutes later, for a total closure of the airport of about one hour. However it seems this will likely only be the case for the first few landings, as elsewhere in the same document it states:
Due to the approach from the west to a landing on the Runway 18L-36R, the operation also has the
potential to affect aviation operations using Runway 18R-36L. Sierra Space’s Concept of Operations would
indicate a time period during which Runway 18R-36L may be unavailable to landing traffic as the re-entry
vehicle transits to the extended Runway 18L-36R centerline. Concurrence with the FAA ATC would be
required to determine the amount of time Sierra Space would need to ensure the approach corridor was
free from conflicting traffic.
This passage would only make sense if the expectation is that after the first few flights runway 18R/36L will remain open during the entire reentry and landing procedure, unless Dream Chaser is on a trajectory that would cause it to pass through or near the 18R/36L approach or departure corridors.
As mentioned the clearances and procedures for suborbital point-to-point landings would likely be similar to the procedures for the Dream Chaser orbital landings.
I will give an explanation now about my statement that Virgin Galactic is not currently, or for the foreseeable future, capable of operating point-to-point suborbital flights. I am looking at it from the requirement that sustainable point-to-point rocket travel would have to be a practical alternative to airline travel. For this discussion I am assuming that price is not a factor. Virgin Galactic’s SpaceShipTwo (and upcoming SpaceShip III) are not really up to this task. SpaceShipTwo is released about 25 miles (41 km) from the landing site at Spaceport America in New Mexico, following a one hour climb to the 45,000 ft release altitude while attached to the White Knight Two carrier plane. After release from White Knight Two and igniting its rocket motor, SpaceShipTwo climbs nearly vertically to an altitude of about 55 miles (300,000 feet), then literally falls back down into the atmosphere and glides back to the airport for a landing.
A more horizontal flight path could carry SpaceShipTwo farther, as speculated in this question. However the answer given there of 3,000 km (1,800 miles) seems way too high. My estimate would be that SpaceShipTwo could go perhaps 40 miles downrange during the one minute of powered flight. That's pretty much a guess, but it's based on the assumption that the observed reentry speed of about 2,000 mph is the maximum speed they can fly laterally. They could climb higher and go say 3,000 mph but then that's how fast they would hit the atmosphere on the way down and I tend to doubt that it can take that much heat. 2,000 mph for one minute is about 40 miles. That would be followed by a glide of maybe 75 miles, figuring gliding from 80,000 feet at a 5 to 1 glide ratio (only number I could find). This would be in addition to the 300 miles or so that White Knight Two would carry it downrange prior to the release (observed average speed of 260 knots for one hour, 35 minutes of which is spent climbing to altitude). For a total of about 400 miles.
If that estimate is accurate that is similar to relatively close destinations such as between Los Angeles and San Francisco which are 350 miles apart, and leaves some margin. That's great, however a subsonic airliner can make the same LA to San Francisco flight in an hour and a half, so this would make it more of a novelty flight since the Virgin Galactic flight including the White Knight portion would take almost as long.
Longer suborbital flights would need more rocket power and it would also generate a lot more heat during reentry. There are no known current plans for Virgin Galactic to develop this type of vehicle, even if Richard Branson at various times has stated some grand future plans. And coast to coast or intercontinental point-to-point flights would require essentially orbital class rockets and reentry vehicles. This is why you sometimes hear the SpaceX Starship mentioned as a possible point-to-point vehicle.
But your question about landing at a commercial airport is still relevant even for Virgin Galactic, who could benefit from offering suborbital tourist flights from airports around the country or even around the world. Although more likely these flights would be conducted at regional airports located near major cities that have long enough runways, for example Palmdale Airport which is located 40 miles from Los Angeles.
One noticeable difference between Dream Chaser's orbital reentry compared to Virgin Galactic's suborbital hops, is that VG's coordination with ATC would revolve around the release from White Knight Two, since that starts the clock ticking about 15 minutes prior to landing. This is compared to about 45 minutes prior to landing for a Dream Chaser deorbit burn. Each has its advantages as it relates to coordinating with an airport. The procedure would probably be that the Virgin Galactic pilots would inform ATC of the planned release time and estimated touchdown time. ATC would then approve or deny the request based on the expected traffic situation at the planned landing time. If the request is denied, White Knight Two could simply do another lap in its racetrack pattern at 45,000 feet, and then make another request with updated release and landing times. Although at the less busy airports that Virgin Galactic would likely be conducting the tourist flights out of, I would think that in most cases the initial request would likely be approved, especially since I'm sure Virgin Galactic would be coordinating with the local airports prior to each flight.