She was in turn filmed by others, so today we have proof of her presence on the square (such as this photograph), but we don't know who she was and why she was filming the soon-to-be-dead president.
She never came forward. Police and FBI did not find her, and the film shot from her position never turned up.
In 1970 a woman called Beverly Oliver claimed to be the Babushka Lady, but she wasn't able to provide the convincing proof (in the form of her film) and so has been dismissed as an impostor.