Do the J2ME CLDC APIs support JDBC based database connectivity?

No, not built-in, anyway. You would have to use some other mechanism (say, a Socket plus some of your own logic) to open a connection from the device to a proxy somewhere, which could use JDBC itself to access the database.

Alternatively, PersonalJava-based devices may optionally implement JDBC in the device. Please refer to the FAQ entry "What is PersonalJava?" for more information on PersonalJava and the J2ME Personal Profile.