I did some research and “accidentally” seems to be the proper spelling. However, some discussions mention that “accidently” is acceptable in informal writing and it is used quite frequently.
The reason “accidently” has become popular seems to be that many people pronounce the word that way.
So is “accidently” just wrong? Is it treated differently between GB/US English?
Accidently has an entry in the (full) OED, but it’s labelled as non-standard:
Now nonstandard. In an accidental or chance manner; accidentally, by chance (contextually: by mischance). (In modern use only as an occasional spelling variant of accidentally adv.)
Interestingly, the first citation for accidently is (tentatively) dated 1425, only a few decades after the first appearance of accidentally (1398).
Merriam-Webster also includes accidently, which it accepts as a variant spelling: