"My husband is executor of a will with his elder sister but has no access to the house key and garage. Is he allowed to the have the keys to the estate the day of his mothers death? The sister is renting the house to her daughter. Is that a problem?"
Yes, an executor is entitled to have the keys to the house as soon as the deceased has passed away. The executor's job at this point is to identify and protect all of the assets of the deceased. He's responsible for any financial loss to the estate, and it's pretty hard to control the assets if he doesn't have access to them.
Is renting the house to a family member a problem? Well, that depends. Is the rent being paid into the estate? If so, and if it's a reasonable rent (i.e. that they would get from any tenant on the open market), then that's probably a good use of the house until it's sold or transferred. The executors are supposed to maximize an estate, so if they can get a few months' rent out of a property that would otherwise be sitting vacant, that can be a good idea.
Your husband's sister will have to realize that when the house is sold, her daughter will have to move out, even if she doesn't really want to, and the executors are the ones who have to enforce this. The sister can't delay the estate just to make sure her daughter is happy.
Is there fire insurance on the house? The executors had better make sure that there is, because if it burns down while still in the estate name or the deceased's name, they are on the hook personally for replacing it.
The two executors also need to make sure that nothing goes missing from the house. The household furniture and decorations, and the deceased's personal items (clothing, jewelry, photo albums, etc) are all part of the estate and the executors might end up paying out of their own pockets if things disappear. I'm not suggesting that the daughter might steal them; I'm suggesting that an awful lot of people seem to think that when a parent dies, it's open season on personal goods and they tend to get taken away as mementos by people who are not entitled to have them.
To me, the problem is not so much that the house has been rented, or even that the tenant is a family member. The problem is that the executors aren't working together. This isn't too surprising, as most siblings don't seem to work well together as executors, but it could lead to trouble if the sister continues to take steps without involving your husband.
Please urge your husband to speak to his sister about arranging a time to spend a couple of hours talking about the estate, just the two of them. They should be sitting down together to figure out steps to take, a timeline, and a list of tasks. If neither of them really knows what to do, they should call a lawyer or trust company to ask for guidance.