Hi, Michael!
In address books of big cities and other sources I saw 6 spellings: with double 'pp' / single 'p' and with vowels 'o', 'a', and 'e' : Rappoport, Rappaport, Rappeport, and Rapoport, Rapaport, Rapeport. Once and while I saw other variations of spelling.
My grandmother always used the spelling with double 'pp' when asked her maiden name. Accordingly, when I started "investigate" my family history, I looked only for the "right" spelling.
However, as I found pretty soon, in some documents, the last name of my great-grandfather was written Rappoport with the double "pp", but in other documents it was Papoport with single 'p'. The same with his son, my granduncle. This is for the period between 1890-th and 1950-th.
In the family of our relatives, I saw spellings alternating between 'o' and 'a' in the second vowel: Rappoport turned into Rappaport. The husband of my great-great-grandaunt was Rappoport, but in the next generation his son took the the name Rappaport. His daughter's maiden name in her marriage record was written Rappoport, with 'o', but when she was granted French citizenship, in the official publications her maiden name was written Rappaport, with 'a'.
I have a strong impression that in XIX and the first half of XX century people and government authorities did not pay much attention to the variations of spelling. Though in address books and similar they were listed separately, sometimes on different pages.