I’m a December graduate. Do graduate programs accept applications for a January start?

Most do not. The standard admission schedule is to apply in winter and start in the fall (August or September). 

As an alternative, consider looking for a job or internship to occupy the months between your graduation and starting graduate school in the fall. You can also inquire about opportunities to work as a post-baccalaureate researcher at your current or new institution.

Regardless, you should plan to apply on the same schedule as everyone else.