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). It may be possible to defer your start by a few months, though you will miss out on orientation activities and bonding with other new graduate students. 

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.