Assuming (a) no taxpayer money is involved and (b) they disclose that being woke is a condition of admission, they should be able to do anything they want. They can be the Hillsdale College of the left, as far as I would be concerned. BUT, big but, if they receive taxpayer money and if they claim to have academic freedom, they should be held accountable.
If they have students with guaranteed student loans, Pell grants, or GI Bill benefits, if any faculty have received federal grants, if it qualifies as a not for profit, it is under the shadow of all federal regs.
At least that’s the way it is for any instution thought to be harboring conservatives or Christians.
I think nowadays the "Bake the Cake" rule obtains. Latest example: In NY they're saying you have to do African-American styles or you're discriminating. Open to the public means all the public. This isn't your grandfather's free country anymore.