Can I place photographs I've taken of vintage cars on mugs, t-shirts, etc for sale on the internet?

Yes, if you took the photos, then you own both the photos and the copyrights to the photos, and you can exploit them in any merchandise you want. The owners of the cars have no rights to the photos, so you don't need their permission.

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.