As Mr. Rewrite suspected, it's driver's licenses, with the singular possessive carrying over to the plural form. It just makes sense, doesn't it? But he can't figure out why this was so difficult to confirm outside of the AP Stylebook.
The dictionary should have sorted this one out. But Mr. Rewrite's paperback New World doesn't even have an entry for driver's license. Merriam-Webster Online or Dictionary.com appear to punt on this issue, carrying only the singular form.
Mr. Rewrite was frustrated initially when the online AP Stylebook returned nothing on searches for driver's license and just driver's. But a search for license smoked out the entry for driver's license(s).
For the record, The Elephants of Style notes that things change when the context splits paired words such as driver's license, as in: After pulling over 24 cars smelling of grain alcohol, police suspended all of the drivers' licenses.
There seems to be quite a bit of confusion over this point among news organizations and would-be news organizations. Google News has scores of articles referring to drivers/drivers' licenses, many from outlets that employ editors.
Mr. Rewrite just remembered that he has to get some traveler's checks. Bye for now.