Can a renter just provide driver's license # and not social security # when trying to rent an apartment?

The DL may not be genuine. His SSN is essential so you can check his tenant history, his employment and criminal background, or sue him or find him. You should refuse to rent to him on the basis you cannot check him out. Maybe he is a spy from the Fair Housing agency so make sure you are legally covered when you refuse his application if he is an ethnic minority.