Your basic headshot? Yes. If you're the only one being asked for a photo? No. If the picture requires nudity? No. There are probably other exceptions also, but this isn't an unreasonable request for your employer to make.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Yes. Your employer can impose reasonable conditions of employment. Many companies have ID badges, which require taking a photo, or your employer may just want a photo so they can remember you later. (Large companies or businesses with high turnover, like restaurants, may have trouble remembering people and a picture might help.)