I think all mean the same thing

I got accepted to Yale university

I got accepted into Yale university

I got accepted at Yale university

This is just IMO...

It's subtle, but I think "accepted into" refers to a group specifically comprised of people.

In each example below, I think statement (a) works best.

I think statement (b) would imply something other than the right to be admitted. I'd interpret statement (b) as a reflection of the hospitability or friendliness encountered there.

Finally, I think statement (c) would be more appropriate if you replaced "chess club" - a specific group of people - with an institution, such as "Yale University", which would consist not only of people, but laboratories, athletic arenas, libraries, and myriad other features which comprise a university.

(a) I was accepted into the chess club.
(b) I was accepted at the chess club
(c) I was accepted to the chess club

(a) I was accepted into Congress
(b) I was accepted at Congress
(c) I was accepted to Congress