"informer," 1785, probably from underworld slang meaning "the nose" (1700), which apparently developed from an earlier meaning "fillip on the nose" (1670s). Snitcher in same sense is from 1827.
1803, "to inform," from snitch (n.). Meaning "to steal, pilfer" is attested from 1904, perhaps a variant of snatch (v.). Related: Snitched; snitching.
Someone snitched my wallet!