I know that we can re-fuel fighter jets mid-flight, but do we have also a mechanism to reload them with ammunition during mid-flight so they don't need to go back to base to reload?
Short answer: no.
Longer answer: you would need to design a carrier craft that could store large quantities of ammunition in safe conditions - enough to supply more than one fighter, perhaps several times. Having a lot of live ammo on board would increase the chances of something going off at a bad time.
To continue: this ammunition then needs to be transferred to the fighter, while in flight at speeds of what, say 200 kts? How do we do this? Perhaps some sort of pneumatic tube would work, but only for the small stuff and you would have to make sure it would arrive safely. Transferring whole cassettes of rounds or bombs would need some other means.
OK, so let's say we transfer the ammo with some sort of failsafe on (pins, etc.). So now the fighter pilot needs some means of removing the failsafe in order to use the stuff.
Perhaps the only feasible solution would be a carrier aircraft that could take the fighter on board while refuelling and rearming. That is what they built the Akron zeppelin for in the 1930s (Wikipedia link: https://en.wikipedia.org/wiki/USS_Akron_%28ZRS-4%29). Don't think it would be feasible with modern fighters, though.
Aerial rearming has been a daydream of military tacticians for decades. However, the task is much more technically demanding than aerial refueling, which is in itself a fairly demanding task for pilot and boom operator just to get a 50-foot fuel hose hooked up.
The technical issues include:
Physically capturing and docking a smaller aircraft with a larger one. The USAF tried it back in the 50s with a B-36 carrying a specially-modified RF-84F. In those days there were no computerized flight controls, and docking the fighter back up with the mothership after it had done its thing proved to be more dangerous to the bomber than just running the mission without a fighter escort at all. It would not be inconceivable for modern avionics to improve on that, however the relative size of fighters compared to larger support craft has increased considerably. Here is a KC-135, built on the "Dash-80" airframe that also sired the 707, refueling an F-15C that's not really that much smaller when you look at it:
Placement of most ordinance under the aircraft. Most fighters and ground attack aircraft carry weapons on external hardpoints under the wings (the RAF-variant SEPECAT Jaguar being a notable example of both overwing pylons and the insanity of British aircraft designers):
When in refueling position under a tanker, the aircraft's most deadly weapons are on the wrong side of the jet to try to get to them to replace or reload them. You'd have to either bring the aircraft inside the mothership, or have it land on top (and provide a sheltered work area for a crew to get to the ordinance pylons).
Relative fuselage width of mothership versus wingspan of fighter. The largest fixed-wing craft in the U.S. arsenal is the C-5 Galaxy. It's a big plane (though not the biggest, that distinction belongs to the Ukrainians with the An-225). Here it is with a couple F-15s and a C-130 parked nearby for scale:
Even with a plane this size, able to transport other planes, you can see from the below photo that neither the F-14 being loaded nor the Galaxy is in a flight-ready condition at the time:
I'd be interested to see whether one of the light weight fighters (F-16, F/A-18) might fit with less disassembly (the F-18 might if you folded the wings), but it remains that you're just not going to be able to fly a modern fighter into another aircraft in midair, at least not with any currently-flying airframe combination.
Getting crew access to the plane. If you've ever seen a ground crew loading a fighter for a mission, you know it's a fairly complex process of not only putting the ordinance (which can weigh up to 2500 pounds for large laser-guided bombs) onto the hardpoints, but wiring up the arming and firing circuits for each one, and programming the plane's FCC with the loadout. This requires getting a lot of people underneath the aircraft (it can take two or three people to load one hardpoint especially if the ordinance has to be muscled into place; here's two people loading probably the lightest piece of external ordinance the USAF has onto an F-15C, and it's only two people because they're using a hydraulic loader):
As the aircraft can't be taken inside the mothership (see previous point), the only other possible place for this to be done is on top of the mothership. That means you'd have a crew out on top of the aircraft muscling ordinance into place. That first requires a sheltered area on top of the mothership that the fighter can be docked and tethered to, as there's no way a crew is getting anything done in a 150-knot headwind. The fuselage vs wingspan difference is important here too; you're not going to be loading AIM-9s onto the wingtips of an F-16 or F-18 when the F-16's wingspan is 33 feet while the cargo deck width (a stand-in for the amount of usable fuselage width) is only 19 feet.
In summary, loading/reloading an aircraft is a serious operation requiring a dozen people to do quickly and efficiently. Putting those dozen people, and their equipment, and all the ordinance into a mothership, getting the fighter to safely approach that mothership and touch down on top of it, then getting the crew into position to remove expended hardpoint pylons and reload new ones, then getting the plane to safely get back off the mothership... It's just not gonna happen with any current technology.
There was a proposed design for a twin-fuselage C-5 as a shuttle transport aircraft; the idea was that this aircraft could be used instead of the risky crane system to pick the shuttle off the ground at an alternate shuttle landing site like Edwards AFB:
An aircraft of this scale might, might, be able to support an aircraft coming in between the tails, docking, then using remote arms to load ordinance. This aircraft would, by wingspan, be the largest plane ever to fly, and would compete closely with the An-225 on every other size stat except total length.
In short: No, there is absolutely no feasible way of integrating midair rearming with our current equipment. This would require a retooling of every pylon in existence to fix a problem that, frankly, doesn't exist.
The actual long answer is far outside the scope of this question and involves explaining the intricacies of the pylons to include thing like arming wires, sway braces, and a whole litany of other insurmountable problems that current technology cannot address. And that's if you could even find a reliable method to transfer a 500 pound piece of concrete from one aircraft to another. Hit any turbulence and that bomb becomes an inert wrecking ball that is sure to do damage in the 100,000's.
Others have clearly described why this isn't done today. But we should be clear about why aerial rearming is such a big deal and why it is likely to happen eventually. A drone airforce has its weakest link in the airstrips or aircraft carriers that it requires to operate. They have to be close enough to the battlefield to allow sorties to be rapid. Currently aircraft carriers are a primary solution to this problem, but it is no secret that they are extremely vulnerable to anti-ship missiles. A drone airforce that could automatically transfer fuel and ammunition becomes very powerful and scary. A swarm of attack drones with tending drones that can operate within 100 miles of the battlefield become much more powerful and flexible compared with an aircraft carriers who must stay in deep water and whose stand-off is increasing steadily as anti-ship missiles improve. Imagine transport and refueling aircraft that could replenish the tenders, and you have a fully automated supply line that can extend anywhere around the globe. What is more, it is quite survivable because each piece has a backup. You would have to take out many transport or refueling aircraft rather than one or two aircraft carriers to take out the system. Science fiction movies have used this arrangement for a long time. Earlier answers have explained why this is very hard using existing technology. But with Tesla building systems that can automatically replace a battery on a car, it should be possible in the not so distant future.