How do I go about working for AP as a photographer?

I would suggest going to school for photo-journalism first, unless you’re already an established/professional photographer?

I imagine a lot of their photos come freelancers.

From the AP’s website: FOR PICTURE PEOPLE
Headquartered in New York, but with bureaus across the world, AP photographers and editors work around the clock, combining the finest in talent with the most advanced photo technology. With 30 Pulitzer Prizes for photography, we offer opportunities and challenges far beyond those of any other news organization.

Also: https://careers.ap.org/index.html

I would start in newspapers before working your way up to AP.

Network. See if you can find someone who is an AP photographer. Talk to them. Find other reporters. Ask how they got into the business. Ask them for advice about how to get into the business. Tell them how much you love their work. Ask for other names

Offer to volunteer for one of them, or ask them to see if there are volunteer opportunities where they work. Then see if you can do a good job and turn it into a paying job.

Just start sending them your stuff.