Just because you request an ADA accommodation, doesn't mean you will be granted one. The standard is that the accommodation must be "reasonable" or something like that. I don't know what your job title is, but private office? Maybe if you are a manager or highly valued tech person, a doctor, or other professional. But if your job is educational assistant or call center operator, that is never going to happen. All HR or the company has to do is say "sorry, no rooms are available" or something like that.
Then your next step is to seek legal redress. Are you willing to go that route? Can you afford it? Do you want to make that much of a stink at your workplace?
Retaliation is also a possibility.
I'd say go in politely and humbly. Good luck. I was granted some accommodations, but it was for a very visible physical disability, which I think gets better treatment.
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A finger in every pie.