Is it possible to get permission for a GA flight to land at Reagan National Airport?
The TSA operates the DCA Access Standard Security Program (DASSP), which allows general aviation flights originating at designated gateway airports to fly into DCA, however there are many restrictions in place, some of the most significant being:


You will need an FAA "reservation slot" to fly in to DCA
(This is required for all unscheduled flights into DCA)
You will need to submit a flight manifest &amp; undergo TSA passenger/baggage screening
The flight crew will need to be cleared by the TSA (fingerprint/background check)
You will need to carry an Armed Security Officer onboard


So the answer is "Yes, you can do it, but you probably don't want to."
It also makes no sense to do it in something like a Cessna 172 as you're giving up at least one seat for the security officer. The program is really geared toward larger aircraft (business jets, etc.) transporting executives with a business reason to be in DC, like testifying before Congress.