The Council of American-Islamic Relations (CAIR) has completed a survey (PDF) of state laws regarding driver's licenses and religious attire. The information was obtained by filing Freedom Of Information Act (FOIA) requests or by writing to the government agencies in each state responsible for driver's licenses.
The survey covers the issue of whether people are permitted to wear headgear in driver's license photos, if exemptions are permitted as religious accommodation, if non-photo driver's licenses are available, and how face veils are handled.
As the report notes, and as I've noted previously in this space, this is not a question that solely affects Muslims. Many Jewish men wear yarmulkes; Sikh men wear turbans. Jewish and Christian women may wear religiously-prescribed headcoverings. Additionally, a number of Christian groups (especially the Amish) have religious prohibitions against the taking of photographs. Most of the case law on driver's license photos in fact has to do with Amish or other Christian groups who don't want photos to be taken at all.
On the specific issue of face veils, three states have what I consider the most reasonable policy, allowing face veiled women to get a non-photo driver's license. These states are Kansas, Pennsylvania, and Indiana. As the survey notes, these states have large populations of Amish and other groups like the related Mennonites who have religious objections to all photos; that appears to be why they're so accommodating. My own state, Washington, allows photos in which the person has their face veiled, but then a notation is put on the license that it doesn't serve as ID. My position is, and always has been, that a face-veiled woman should be able to obtain a non-photo license as long as she is qualified to drive, and that she should obtain a separate photo ID for times when her identity needs to be verified. In a situation where it's only her authorization to drive that's at issue and not her identity, she doesn't have to display the photo unnecessarily.
Anyway, the survey is a very good overview of the law in the different states and should be a good reference for those interested in this topic.
Added: The survey led to a news story from Kentucky about its driver's license laws.