Factual error: During the orphans' rendition of "You're Never Fully Dressed Without a Smile" Molly is seen with a plastic drink bottle (that makes a plastic-bottle-esque "thunk" when thrown). The movie takes place in 1933, however polyethylene terephthalate (the kind of plastic used for drink bottles) was not developed until 1941.
Factual error: They take Annie to the movies to see Camille except there's a problem - Annie takes place in 1933 and the version of Camille they're watching (starring Greta Garbo) didn't come out until 1936.
Factual error: Throughout the movie, all the women (with the exception of Miss Hannigan) are shown to be wearing full coverage pantyhose (you can most see this in dance numbers). Pantyhose that covered all of the leg and buttocks were not invented until 1940 (Annie is set in the 30s). Any hosiery worn prior to that would have been stocking type that only came to the top of the knee.
Answer: Mrs. Hannigan's box of Annie's parents' things might have the only record of her birth certificate. This is the 1930s, after all. Paper records went missing all the time.
Captain Defenestrator