If you are in California
http://www.dmv.ca.gov/dl/dl_info.htm#SSN
The above link sounds like you don't need to present the SSN (as you cannot have one for H-4). However bring your passport, I-797 form (if available) would be useful in DMV.
If not in California, as always, Google is your friend, search with "H-4" and "driver license" would give you pretty good ideas for the process I believe.