I'd call in advance, but the FBOs should be able to shuttle you down there or let you take a crew car. There's also a Loudoun Transit bus from the main terminal to the museum and back.
Originally, they were going to allow you to taxi down to the museum. We met with the A&S Society people at Oshkosh about this (we were already based at IAD at the time). That sort of went the way side after 9/11. To get the gates open between the airport complex and the museum takes a lot of coordination between the Smithsonian, MWAA, TSA, and the FAA. My wife did manage to convince the museum to host at least one event a year (Originally BecomeAPilot day, later Innovations in Flight). It let 50 or so small GA planes in along with various military stuff and for the past few years a United big-assed jet.