I'm not in public relations. I'm going out on a limb and claiming that, even as a complete amateur, I could help some organizations avoid pitfalls.

I know I could have helped Overlake Medical Center.

Washington state hospital apologizes after inviting donors to sign up for Covid-19 vaccine

We all know that everything favors the wealthy. If you've got money, everything is faster and easier. I doubt anyone expects that to change. However, the PR playbook suggests organizations should go to great lengths to (1 - best case) avoid enabling those wealth-related disparities or (2 - I'm on my way to being a cynical PR fixer case) avoid getting caught enabling those wealth-related disparities.

Bellevue's Overlake Medical Center should hire a PR firm with a Napoleon's Corporal. (Or skip the PR firm and get some solid second opinions.)

Overlake Medical Center: Hey, we have some unexpected slots open for COVID vaccines. We haven't got a mechanism in place to make sure the vaccines get used. We're thinking about sending out an email to let people know.
Napoleon's Corporal: That makes a lot of sense. The more jabs in arms the better. Who are you sending the email to?
Overlake Medical Center: First responders, volunteers...
Napoleon's Corporal: Excellent!
Overlake Medical Center: ...and major donors.
Napoleon's Corporal: Eeewwww.
Overlake Medical Center: That's a bad idea?
Napoleon's Corporal: It's somewhere on the Mt. Everest of bad ideas. It's not up there with invading Iraq or waterboarding prisoners, but it's on the same mountain. There's a global, deadly pandemic that is disproportionately affecting poor people who are out of work, live in small homes, and struggle to pay for health care in the best of times. Your wealthy donors live in Medina in 2,000 square foot homes with backyards. One of their top pandemic complaints is that they've run out of new breads to bake.
Overlake Medical Center: Ahh, fair point. We just want to make sure the vaccines get used.
Napoleon's Corporal: That's commendable. A wasted vaccine doesn't benefit anyone. But I'm sure we can look at a few different options here...
That's a nice thought. But here's how Napoleon's Corporal would really play it.
Napoleon's Corporal: Do not mention donors. Do a quick database query. Find your big donors who meet the Phase X requirements. Send them a generic email. Hello, we hope you're doing well in these trying times. We're trying to make our vaccine system as efficient as possible. We've noticed that you qualify for the vaccine. We have appointments available. If you live in a multi-generational home (wink wink nod nod), please sign those people up, too.

If any hospitals want to hire a blogger who occasionally offers PR advice, send me a note. I guarantee I'll always steer you away from (blatantly) giving the impression that healthcare is influenced by net worth. You know, it's all about the wink wink nod nod.