You don’t have to design the trial that way to assess if vitamin supplements improve the treatment outcome. You would just have 2 treatment arms: standard of care (SOC) and vitamins + SOC. This is how many cancer trials are set up. You could also have a vitamin only arm since untreated ADHD isn’t a death sentence. You don’t learn if vitamin insufficiency causes ADHD, but you can at least learn if it treats it.

There’s a big problem with your proposal for discovering if vitamin deficiencies cause ADHD, though. Even if you show a causal link, it doesn’t mean that’s the only cause, or even the proximal cause, of ADHD. For instance, you might establish that a lack of cobalt in the diet is linked to ADHD symptoms, but then find that in the ADHD patient population that cobalt deficiency is not detected. What’s going on? Likely, cobalt deficiency is an upstream trigger for a common pathway to ADHD symptoms that also has multiple other endogenous and exogenous triggers. So, not only does a trial design like your example put the participants at very high risk for poor health outcomes, it also likely leads to no actionable results. This is what the IRB is going to cite when rejecting the proposal. (NB: cobalt relationship with ADHD is just a placeholder example, not a hypothesis or statement of fact).