Yes, you can. A Drivers or Learners "Permit" is permission to drive and therefore a temporary license with certain restrictions.
You can obtain your own auto insurance or you can be covered under someone else's policy such as your parents or the vehicle owners policy, but you must have coverage. If you are a minor and want your own policy, due to the limitations of contract law your parent or legal guardian will need to countersign your insurance application for it to be legally binding.
A Drivers Permit comes with all the responsibilities of anyone licensed or not who operates a motor vehicle on public roads, including our financial responsibility. No state provides exemption from Financial Responsibility Laws for minors or permitted drivers.
Happy Motoring
You can get insurance with a permit. I live in Florida and got it with a Learners Permit
Some Insurance Companies may not require you be added to the vehicle owners or parents policy until you complete your test and get your drivers license, but you can't drive uninsured in any event. It's best to contact your Insurance Company and ask what the policy is for newly permitted drivers.