Ultimately, you'd need a very large sample size or ensure you have an even-ish number of all nationalities/peoples/ethnicities tested for this autoimmune response.
That's the whole point of random sampling/random assignment. You don't even have to plan out all of the things you want to control for (nor in fact SHOULD you even, many would argue).
You merely choose several research sites in major regions of the U.S. and go to town. And yes, you then have to run A LOT of participants, as you say. The 100,000 I mentioned is probably much less than you would want to run. But please note that that's already 20x more people than what they do currently. As you run many people, the black and white people and the people with different genetic dispositions will naturally end up closer and closer to 50/50 in your test cases. Just like flipping a coin a lot will get you more reliably closer to 50/50 on average.
But hell, if you want to, you could run
6,000,000 participants, only half of which are in a no-vaccinate condition, and so it would only lower the vaccination coverage by 0.9% (10% of them already would have chosen to not vaccinate as-is if not randomly assigned to that condition), and even then, it would only lower it for the duration of the study, after which the other half could get the vaccine if they wanted to. I.e. in a year or so maybe. That's not a big deal by any stretch of the imagination.
You're only actually altering your country's lifetime vaccination coverage by 0.9% / 78yrs = 0.012% by running that 6 million person studyAnd that's just the U.S. Collaborate with some other nations if necessary and you can do way mroe than that, without any of you having to meaningfully alter your vaccination coverages.