Yes, they get a high school diploma. How? Issued by the parent who has complied with state educational laws. Homeschooling is legal in all 50 states as a form of k12 education.
Homeschool students apply to colleges and universities across the country. They are accepted to elite schools, earn competitive scholarships, etc. I have been homeschooling since the early 90s. I have a ds who is a chemE, a current college student who will be applying to grad school the fall for physics, a rising college freshman who was awarded a university’s competitive scholarship, etc. I have friends whose kids have graduated from their homeschool and attended Stanford, MIT, Berkeley, Princeton. I know homeschooled class of 2017 grads who were awarded Vandy’s Cornelius, W&M’s 1693, a full-ride scholarship to Duke, etc.
Students can study AP courses at home, enroll in AP courses online, enroll in local homeschool AP courses, etc. Parents can submit AP syllabi to College Board for approval and can label courses AP on transcripts.
Some examples of online courses are
http://www.aphomeschoolers.com
https://ohs.stanford.edu/
Students do not have to enroll through a designated program. They can study homemade courses. They can study via courses like MIT’s opencourseware, etc. There are as many ways to homeschool as there are homeschool families. (I can pretty much state unequivocally that my homeschool is not anything like the majority of the posters on this forum. My homeschool does not resemble a school at home environment. My kids take very untraditional type courses. It works for us.)