I still think a better strategy would be to focus on taxis, for the reason that they are commercial vehicles and as such can be regulated. Plus, taxis have been losing a lot of marketshare to Uber and others, and if it were presented to them as a way to regain customers, they might be more receptive to it. I just don't see most Uber drivers, who are currently paid very little once expenses are accounted for, being willing to not use air fresheners, perfume, aftershave, scented laundry detergent, scented shampoo/conditioner/soap, etc... for a couple bucks here and there.
MCS is a really huge problem for a relatively small number of people. I'm not saying this to dismiss anyone, but we need to be realistic here. For people with MCS, I know it's truly awful, but I think when trying to make changes it is really important to have a strategy that has a reasonable expectation of working. I think in the unlikely event Uber did have a rule, it would be extremely difficult if not impossible to enforce, and would probably result in a LOT of drivers walking away from the company...which Uber might not be willing to risk since they are hemorrhaging money like crazy and are posting losses year after year. I don't think they'd be keen to jeopardize availability of drivers.
What might work instead of a petition asking for an unenforcible across-the-board ban would be to ask Uber to promote the idea that specific drivers could choose to be chemical-free, or to have some kind of option on the app so that drivers can request a chemical-free driver, in exchange for a small surcharge. Once there is a financial incentive, I bet more and more drivers might be willing to go this route. In general, when trying to get a person or company to change a policy, it's always a good idea to present concrete actionable things they can do that are mutually beneficial. I'm afraid that just signing a petition like this is a real waste of time that could be better spent.