Can they check themselves into a nursing home?

It's not a hotel. Your mom can't just walk up to the front desk and request a room. She would have to meet with the director, take a tour, and then there's the financing. Your mom will have to fill out forms regarding her finances and agree to sign over her social security to the nursing home each month. If your mom can't afford this she will be advised to apply for Medicaid. This doesn't happen in one day.

Do you think your mom will go through this process because she's in a snit? She thinks her life is out of her control now? Wait until she gets a look at institutionalized living.

And since you have taken her car away, how does your mom expect to get to the nursing home?

My aunt did that. She walked into her local care center, where she had been an LPN, and said she thought it was time to check herself in. All of the paperwork got done and she lived there the rest of her life. She was not in a snit or in a hurry. She was just practical.

BUT in this case I doubt that Mom is serious about this. And if she is, and she qualifies for nursing home care, that wouldn't be the end of the world. Stay strong about the car.

We got the same threat, mom said she would rather die than give up her car. She said she would check into a nursing home and be dead in six months. It's just a tantrum and you should ignore it like any other tantrum.
Any nursing home or assisted living facility will evaluate her health, get paperwork from her MD and ask for her financial records to find out if she can pay. Take her on tours of nearby facilities.