Is the at fault party's insurance company required to provide rental vehicles if the injured party's car was destroyed?

That depends on the kind of policies in place at the time. Ask your agent to help on this question. Or, your lawyer should make a demand.

You need to distinguish the responsible party from his insurance company. The responsible party is liable for all damages suffered as a result of his negligence. His insurance company will always, I mean ALWAYS, try to pay less than that. You, however, are not obligated to "take it or leave it". If you are not satisfied with the offer, then your recourse is to sue the owner and driver of the responsible vehicle and let a judge or jury decide all to which you are entitled. They do so by their judgment award. If your vehicle which was destroyed was handicapped equipped, then you are entitled to a replacement for a similar vehicle. You would also be entitled to a similar vehicle, on a rental basis, until you obtain a replacement vehicle. You will need documentation to prove what kind of equipment your vehicle had.

It depends on the wording of the policy. Most insurance companies do this however.

Not under Michigan law.

If they agree that the accident was the fault of their insured, then they are obligated to provide a rental to the injured party, if their policy with their insured includes such coverage.

Not in Michigan.

The at fault driver is liable for all your damages reasonably related to your wreck. That would include a rental for a reasonable time period after the wreck. Usually, that time period is up to the time the property damage claim is either settled or it is determined that it cant be settled.

Not a legal requirement but you should ask for it and the insurers generally help out with this.