Sister refuses to place Mom in a home. Mom is 100 yo and lives by herself. What do we do?

Litigation will not work, will cause more dissention and will be enormously costly. If your Sister is the POA or Guardian for her Mom, as well as caring for her in the home, there would be no way to convince a court she cannot do this; a huge fight could result in the state taking guardianship, which would lead to no one having any say about Mom's placement or her finances.
I think, until your sister is willing to put Mom in care there is little that you can do. She knows the choices. Continue in caring or stop.The family is killing themselves in enabling this situation by being there for her.
A poster recently told us that being a caregiver can kill you, and told us of her own illness and its costs. This isn't unusual.
I am so very very sorry. You simply cannot change the choices of other people. Constantly fighting her pushes her into her corner. Next time she tells you "I cannot do this anymore" tell her that when she is ready to consider change you will all call the family together to hash out what the options are. You must feel utterly helpless, but trust me, sinking 1,000s of dollars and escalating this situation is unlikely to help.

Listen siblings: one person cannot do it all!! Get your backsides there & help the one who got stuck with this awful burden! I speak from experience: ALL of a parent’s children need to pitch in.

I would just call Adult Protective Services and let them do their job.

Tell your sister in a non threatening manner first that you think mom needs a social worker to help out. Hopefully she'll be agreeable, but even if not, your next step is to contact APS and report a vulnerable adult.

If sister says she can't do it anymore, then explain that the alternative is placement. Please make it very clear that you are not going to take on the burden. Also, that it is time, for safety reasons, to find placement. Simply, the time has come (long since) to deal with this living arrangement rationally and to make the necessary changes.

If Mom is out of money, what litigation is going to force your sister to place her and pay for it?

Give your sister a break, for heaven's sake. With all that on her plate can you seriously blame her for finding it hard to think straight?

So, your sister has stated that she can't do this any more. What would your sister like to happen next? Is anybody in touch with your mother's local social services, including but not limited to Area Agency on Aging or, if necessary, APS?

Most of the things you are concerned about were be very much the same if your mother was in an assisted living or nursing home situation. You don't mention any actual unsafe situations, only things that many people are dealing with due to Covid. In my father's assisted living they have had only a limited period where the communal dining room was open and being used. Most of the time since mid-February meals have been delivered to resident's rooms and they eat alone. If a resident leaves the facility for any reason, including medical appointments, they must quarantine (not leave the room, special precautions by staff, etc) for two full weeks. Now, it is a more supervised environment so your mom wouldn't be able to flood the place but it actually seems that aside from that incident not much would be improved by trying to wrestle your mother into a nursing home. Your sister needs to be more specific as to what "she can't do this anymore" actually means and give specific ideas on what help she needs. Then you can work to figure out how to provide her that help but I don't think that your can litigate your way to a different situation.

The Covid restrictions must have made this situation so much harder. I think many that would have called time on unsustainable situations have been forced to just cary on.

Unfortunately this family seem to have paddled their canoe up the creek. They are now calling out for help, but still paddling in the same direction. OP is not on the boat & cannot steer. In fact, they ignored all directions the OP gave.

The main caregiver sister will need to put down her oars. Get some sort of emergency respite care for Mother while she figures it out.

One temporary pathway I see is if Sister or Mother falls or is ill in any way, the other family members call EMS - instead of rushing over to pick up those oars.

Is there a reason you can't visit there yourself to assess the situation and to sit down with your poor sister. Caring for someone with Parkinson's is a load in itself and caring for a grown man with autism is another job. Plus she's caring for your mom. It's way too much for one person. Your sister may not be forthcoming with details because she is so overburdened that she can't even express what's wrong. What may seem pleasantly demented to you is an incredible burden to the person who is responsible for caring for her. Get there and lend a hand.

Sister lives with mother in mother’s house, together with sister’s husband with Parkinsons and eldest son (aged 50+?) with autism. Is that right? If mother’s house is sold to pay for placement, does sister and family have anywhere to go? Could that be why she is so resistant? If mother has no cash, would Medicaid evict sister who has been caring for so long? If not, does sister know that her home is safe? Lack of information could be underneath some of this.