What to do if my workplace causes me to get sick?

You are describing a very complex situation, so you should consult with an employment law attorney. The basic rule is that if you are injured or sickened by your job, you can get worker's compensation. You may also be entitled to disability, but need to look to the criteria of any disability benefits, including disability insurance provided by your employer, to see whether and to what extend you'd qualify. The problem is, however, that fibromyalgia is a condition where (1) not everyone, including every insurer, for example, uncritically accepts it as being an actual condition; (2) related to the above, I don't believe it's causation is understood that well--which means proving your job caused it may be very difficult; and (3) again related to the above, it generally is a condition where proving even its existence or the extent of disability it causes is not easy--there's nothing concrete or objective to point to, as with, say, silicosis caused by exposure to asbestos particles on the job.

In short, this is likely going to be case where you need expert advice, based  on the specifics of your situation, on whether you likely have a claim; and if you do, you will probably need expert  help to present that claim. That's why you absolutely need, if you want to pursue this, to consult with an employment lawyer, or perhaps a PI attorney with considerable fibromyalgia experience. Good luck.