APPIC is a an organization that is focused on supporting internship and postdoctoral training. APPIC administers the match process (although an outside company actually runs the match numbers). Programs can join APPIC as members, and have to meet certain criteria to be accepted. APA evaluates the internship programs to determine that they meet standards for training set forth by the powers that be. Basically, APA accreditation is what you should be focusing on - that's what you want for licensure.
Although you can get licensed without an APA accredited internship, the burden of proof falls on you to demonstrate that your training met the state's licensure standards. That's a lot of extra work and a huge headache.