First thing first, be very cautious about whom you hand out the ID and SSN and other PII (Personal Identifiable Information).
If you have received and sent the communication in an official channel (company mail ID, for example), then mostly you should be OK. Otherwise, politely request them to send the requirement via an official channel, and supply the information in a way that has traces (like, uploading information to a company portal).
Of course, if you have started working already, this should not matter much, but before joining, practice some caution.
That said, to answer your "question" - there's no harm in following up, get in touch with them and check whether they have received the information you shared and whether they'd need something more from you. Don't forget to thank them in the end.