What to do if I got into a car accident and the company who towed my car won't let me get my house keys and credit cards?

That is your property--they have no right to it.

You can contact the police, explain that they have taken your property without your permission, and ask the police to help you get it back. They should help, in a case like this.

However, if the police refuse to help, claiming, for example, that it is a civil matter, not a criminal one (e.g. they feel, even if incorrectly, that it's a dispute or disagreement as to who has the right to the property, not a crime), you'll have to sue to get your belongings. Since lawsuits can be expensive and can take time--even moving for a temporary restaining order, the fastest form of relief, can take days, or a week or more, you may wish instead to do the following:

1) Cancel your credit card and have a new one reissued;

2) Change the locks on your house;

3) Then sue the towing company for the cost of 1) and 2).

The advantage of doing this is that you should be able to sue for these costs in small claims court, where you can act  as your own attorney. However, if you are trying to get a court order requiring the towing company to turn your belongings over to you, while you are allowed to represent yourself (sue "pro se"), that is a much more complicated proceeding--it can be difficult for a non-lawyer.  So changing the card and locks prevents theft and then presents you with a simpler court case for reimbursement.