There is a Cinderella story from virtually every culture. A girl who does all the dirty work for her step family, via some magical assistance, obtains a dress and means to attend a ball. The prince of course, can not but fall in love at first sight. In most versions she is given away by a shoe that she loses on the stair. It is probably that this story is so old, that it managed to get all around the world and was adapted in many cultures long before such stories were written down. So there is no way to say where the story originally came from, but it's appeal to the entire world is much less of a mystery.
The Cinderella in Into the Woods is clearly taken from the Grimm version, though unlike the versions of the other stories used, this is neither the earliest known version or the most prolific. The earliest documented Cinderella story is Yeh-hsien, circa A.D. 850. The most prolific version around today is Charles Perrault's Cendrillon, which is the earliest use of a name which translates to Cinderella (1697) and was used for the Disney film version. The Cinderella in Into the Woods was first know as Aschenputtel in the Grimm Brothers story from 1812.
The Grimm brothers are the source for many familiar elements. The removal of the toe/heal, the gift giving tree and the pecking out of the eyes all come from Grimm. While most of the other elements are present in all the stories, the golden slipper can be traced back at least to Yeh-hsien.
The absence of a fairy godmother in Into the Woods certainly surprised me at first, but the only version of the story with a fairy godmother was Perrault's. Original elements attributable to Sondheim and Lapine are the character traits there to complicate the story, namely Cindy's indecision and the Prince's hunger for whatever is hard to get.