How do you deal with a mother-in-law who won't talk to you because she imagines you were going to take her car from her?

If she is delusional, she needs intervention for her care. She may just have a UTI and needs to see a doctor and can get antibiotics which will help her, if this is her issue, but it should be ruled out because UTIs are extremely common in women her age and if left untreated can lead to sepsis, which can kill her. If she now needs ongoing care because other medical problems are not causing her paranoia, and no one is her legal PoA or guardian, then you should call APS and report her as a vulnerable adult. They will eventually have guardianship for her and she will receive proper care and be placed into a facility and the guardian will manage all her affairs. Without PoA or guardianship your family has no legal ability to help her. If someone is her PoA this person needs to step in now to help her, no matter what she wants or likes. If she's not in her right mind, the PoA needs to make decisions in her best interests. If at all possible please disable her car so she cannot drive it. Remove the battery or take something else out that disables it. You should also go online and anonymously report her to the DMV and they will send out a letter telling her she needs to come in for a re-test, which she'll probably not do in her current condition. Her license will either expire or be ended (if she goes in for the test and fails). BUT this does not prevent a person with memory and cognitive issues to still get in their car and drive. So please disable the car.

First of all, if she's thinks people are watching her condo and stealing her jewelry and pills, then she has some kind of dementia.
She cannot be living alone anymore.
For the love of God, she CANNOT be allowed to keep a car and drive.
My friend, the problem is not getting your MIL to talk to you again.
Judging from what you're saying here, the problem is that she's out of it and in an unsafe living situation. Please for her sake have a family meeting and either put her in a care facility or make arrangements for her to have homecare if that's possible.

It is impossible to expect that she won’t be delusional while having Alzheimer’s disease.

Other than meds possibly helping her, I have no idea.

She really should be placed somewhere for her own safety.