The stuff I found has all been from FamilySearch, which is also where you go to order (micro-)films to verify baptisms and look for marriages and deaths. Note that this costs money, and it only gets the films to your nearest Mormons (Family History Center, which you have to choose before you order), but it's the only way to access these records in the U.S. (And it's a whole lot cheaper than airfare.)
A quick breakdown of what's available, in general and specifically on FamilySearch:
Civil registration (began Oct. 1, 1895): FS has digitized images for many places in present-day Hungary, but only Szabolcs county and a small portion of Borsod are indexed so far, and privacy laws and other causes mean that coverage is incomplete. If you know where something occurred, and have a pretty good idea of when, you can generally find it by browsing through the records, image by image.
Church records (pre-1895): for places in present-day Hungary, FS has an index of Roman Catholic and Reformed (Calvinist) baptisms, but the actual records themselves are only available on microfilm (not online). For Lutherans and other religions, and for events other than baptisms, it's film only, with nothing indexed.