I know this is a very old thread but I had this problem today and a Google search brought up this forum&thread as the first return.So after fiddling about with my Reaper settings I've found that unless you use an ASIO driver,Reaper won't see your USB mic.Try ASIO4ALL for a free driver of this nature.Then all should be good.
Good luck to anyone looking to solve this problem like me.
RodCow
aka
the high writers
http://www.youtube.com/playlist?list=PL6B5EBABD7FE09ACF