You don't have to worry about this! With pet insurance, you can go to any licensed veterinarian. You pay the veterinarian for services (as you normally would), then submit that bill with a claim form to the pet insurance company. Any reimbursement you're eligible for will be paid directly to you by the pet insurance company.
This is different from human health insurance, where the insurance company pays the doctor directly (not you). That's why for human health insurance, you need to confirm whether your doctor works with a particular insurance company. In this way, pet insurance is simpler than human health insurance.