Can I get a driver's licence without a Social Security number?

Usually yes, but which answer applies depends on why you have no number.

If you are not eligible for an SSN

A number of states issue a licence to residents regardless of immigration status. Where that is the case the licence is normally marked as not valid for federal identification — it lets you drive and buy insurance, and it is not a REAL ID. Whether your state does this is the first thing to check, and the state pages on this site link the relevant page where one exists.

If you are eligible but have not received it yet

This is the more common situation, and the answer is different: most states will accept a letter from the Social Security Administration confirming you are ineligible or that your application is pending, rather than the card itself.

What does not change

Whatever your status, the DMV still wants proof of identity, proof of date of birth, and two documents proving you live in that state. Those are the requirements that trip people up far more often than the SSN question — a phone bill is rejected where an electricity bill is accepted, and a document in a roommate's name is not accepted at all.

Insurance is also unaffected: it is required wherever you drive, and a licence issued without an SSN is a valid licence to insure.