Who can help us resolve a dispute with our property’s management company about required floor repairs in our apartment?

I am puzzled as to how a "corner of a piece of vinyl flooring coming up" evolved into a "safety hazard" requiring "several thousand dollars" of repair over the course of a little more than a year. It really begs the question, how did that happen? If the entire floor was in such bad condition when you moved in, you should've notified the landlord about the general condition of the floor. The landlord now seems to be treating the floor problem as something you created.  Be that as it may, you can still report the condition to the Building Department if you live in a borough of New York City by calling (212)639-9675. If the floor is found to be a building code violation, the landlord will be served with a notice of violation and you can start a housing court proceeding (an HP action) and the landlord will have to fix it. You may even get a rent abatement. If you want the repairs done at no cost to yourselves, you'll need to let in inspectors and repair people regardless of whether they're masked. Since you're immune compromised you can always wear your own masks and ask everyone to keep their distance from you. But you can't force anyone to wear a mask.  You haven't said whether the landlord or super came into your apartment to inspect the floor, or just the flooring company, and whether you or the landlord arranged the company. But I'm guessing that neither the landlord or super has been in your apartment because they won't wear masks. That would be entirely understandable if it were April 2020, but it isn't. If it comes out in the course of a court hearing that you refused to give access for repairs or inspections because workers or inspectors wouldn't wear masks, there's no guarantee a judge wouldn't throw out your case.  You can always stay in another room while repairs are being made or conditions inspected.You can also call around until you find a company or repair person willing to do the repair at a reasonable cost, and pay out of pocket. Just because you were told by one company the repair will cost thousands of dollars doesn't mean it should. As for the landlord's demand for a $2800 deposit, that defies logic. If you choose to hire someone to repair the floor, make sure they're insured and don't alert the landlord.