So I have been accepted to the online program; starting Jan 2021 and will be using the GI Bill.
I am not sure how much I can answer for you, but I'll give you everything I know haha.
I am prior Navy, and I used my GI Bill for my undergrad. My husband transferred his GI Bill to me (he's active duty), so I can use that towards Yale Online. Like you said, there is a tuition cap, however Yale does participate in the Yellow Ribbon program (at least they did last year), which helps cover some of the additional costs of private education that the GI Bill doesn't cover. The amount of additional tuition the program agrees to waive via the Yellow Ribbon Program is not only school specific, but program specific. It is an agreement between the school and the VA. From my own research, I found that Yale agrees to waive $5k per veteran in the PA program that is participating in the Yellow Ribbon. I am not sure if that is per semester or per year. Nor can I confirm that Yale Online is doing the same.
So while the GI Bill will cover a significant amount, I will still pay out of pocket for some of it.
Speaking only from my undergrad/GI Bill experience, I fill out a form with the school's veteran office and they submit it to the VA (I don't remember if this is yearly or per semester). This form essentially states that the VA needs to pay the school directly for said year/semester via my GI Bill. I am assuming I just pay the school directly for the rest of the uncovered tuition.
I am still in the process of getting all my admin stuff set up with Yale Online, so I have not reached the actual payment part (aside from the seat deposit). Please take what I have said with a grain of salt, for my cohort has not even received the final tuition cost yet. I plan on speaking with the Veteran's office at Yale Online to make sure I have all my ducks in a row as well. Again, I have no idea if anything has changed with the Yellow Ribbon program for my cohort or anything like that. This has just been my experience.
Please do not hesitate to pm me if you have any other questions at all.