That is not something I would worry much about. Companies need your SSN for two reasons--first to make sure you are 'real' and not someone they already terminated or someone making up a SSN to 'double dip' shops they are out of rotation for; second to send you a 1099 if you earn more than $600 in fees and bonuses working for them in 2015. You gave them your information, it is now up to them to do whatever verification they want/need to do. There was a phone number my sweetie used to call to verify SSN pretty much on the spot for day workers, so it is probably just a matter of getting around to it. Back when companies did not offer secure websites to enter our personal information, you could enter a SSN of 000-00-0000 and get back with them by phone or email after your first shop to give them the real info.
As a general rule with first shops you should be looking mostly at fee shops rather than starting with a bunch of out-of-pocket expenses to be reimbursed. So yes, it would be grand to be paid for your shops in the normal cycle (and you probably will), but just in case you may want to lay off the big reimbursements, especially if they are going to leave you cash short while waiting for pay + reimbursement.