Judging by the compact regular case, and more generally the spatial case, regular projectivity of locales, resp. regular injectivity of frames, must have something to do with $\neg p\lor\neg\neg p$ and $\neg(x\land y)\to(\neg x\lor\neg y)$. On the other hand, existence of locales without points shows that the terminal locale is not regular projective. I am convinced somebody should already have found out which locales are regular projective, but who?
The same question about projectivity with respect to arbitrary epimorphisms of locales is probably easier but less interesting. Still, I don't know anything about that either.
PS As Simon Henry pointed out, I should rather pick some pullback stable class of locale epimorphisms. I guess I don't know an answer about any of them (except maybe the proper ones), so please just choose an as large as possible class of nicely behaved epimorphisms of your choice - say, quotients, or of effective descent, or triquotients, etc.