Since I got two replies with basically the same thing at the same time, I'll pick one at random and it'll serve as a reply to both. You won the coin flip :)
This is feasible in a sense, but C is a fairly tricky target to compile to: you have to make sure that you don't accidentally include UB in the code you generate. I know pcwalton has lots of feels here...
The easiest way to do it would be if LLVM had a C backend; I know that it did, but it was removed a few years back, and I haven't heard anything about it coming back into tree yet. MIR might also in theory enable new backends, but then you'd have to re-implement all of the optimizations that we currently rely on LLVM supplying.