The answer is: no, normal flights are not allowed under the canyon rim.
If you look at the sectional chart, you see this notice:
Searching through the CFR (Title 14, Part 91) brings up this Special Federal Aviation Regulation No. 50-2 - Special Flight Rules in the Vicinity of the Grand Canyon National Park, AZ.
Ther is also Supbart U of Part 93.
The Subpart U regulations seem to be the most current applicable regulations (please correct me if I am wrong).
The rule applies to everything under 18,000 feet MSL within an area around the park (like the chart says).
Except in an emergency or if otherwise necessary for safety of flight, or unless otherwise authorized by the Flight Standards District Office for a purpose listed in 93.309, no person may operate an aircraft in the Special Flight Rules Area within the following flight-free zones:
It then describes the four zones: Desert View, Bright Angel, Toroweap/Shinumo, Sanup Flight-free Zone. You can't fly through these, but between them are "corridors" in which you are allowed to fly above a certain altitude. See this map for locations of the Flight-Free Zones and corridors.
There are also minimum sector altitudes for different areas, including the corridors, which are going to be the main thing preventing you from flying very low. The altitudes are different for commercial air tours versus transient and general aviation.
no person may operate an aircraft within 500 feet of any terrain or structure located between the north and south rims of the Grand Canyon.
There are also noise limits depending on the aircraft type.
For some info on the background of these rules, see this NPS study.