Under what conditions can a non-RVSM aircraft or pilot fly in RVSM airspace? After purchasing an aircraft, it takes the FAA a long time to issue RVSM approval for the new owner.  

During this time, under what conditions may an aircraft fly in RVSM airspace even though they aren't approved for it?  

What is the process to get ATC approval for such flights?
According to the AIM Section 4-6-8, the language to use with ATC is simply, "Negative RVSM."  You have to say it on the request for an RVSM flight level, flight level change, or climb/descent through RVSM airspace.  You must also say "Negative RVSM" on check-in if currently at an RVSM flight level.

NBAA provides a guide for navigating the RVSM approval process.  This guidance is fairly new, and indicates that approval through a prior operator of the airplane is acceptable.  Your FSDO will make the authorization.

See also FAA RVSM Documentation