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: 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.
Factual error: If Annie was dropped off at the orphanage as a baby and they aren't supposed to have any fun and really don't have any toys or anything, where does she learn how to swim? This is during the depression era. I'm pretty sure these were no community pools. She jumps into Mr. Warbucks' pool the day after she arrives and starts doing a backstroke.