Can car rentals be deducted on my taxes?

Most likely not. Medical expenses have to be for yourself or a legal dependent. I assume that the person is not a legal dependent since he is not a family member. Now, if you were a business providing services to clients, then it would be a deductible business expense.

Ok, well, I signed him up for " In home supportive services," that's how I became his caregiver, so he became my employer. I am paid through this program. Does this still make it so I can't deduct the car rental expenses?

If he is the employer, he can reimburse you the expense, then deduct it from his own taxes if he itemizes. This is for the same reasons -- he's not a dependent and you are not a business if he is your employer. Now if you are self-employed, it is a different matter, because then it would fall into business expenses. However, this goes along with paying your own self-employment and estimated taxes. Not fun.