Yes, in most states if you transport people for money this is considered a livery and you will need to purchase a business or commercial auto insurance policy.
A personal auto policy normally specifically excludes coverage as a livery or delivery vehicle. That means hauling people or stuff around for a fee will not be covered by your personal auto policy (PAP).Since your personally insurance will not cover you when you use your own vehicle if you use it to drive elderly people to their doctor appointments for a fee, thus using if for business purposes, then a "special" insurance in the way of a business/ commercial policy is required usually.
You may ask your current insurance provider if they can offer you commercial coverage for this or if you will need to obtain this elsewhere.
Like your personal car insurance, commercial car insurance is there to protect you and your assets. We can help you find the right commercial coverage for a variety of auto liability, medical, uninsured motorist, collision and comprehensive coverages and deductibles to create a policy that meets your company's needs. You can get a Safeco insurance quote for commercial coverages here.