In all the research that I've done, I could find no auto rental companies that will allow you to leave your personal vehicle on the rental car parking lot.
If you're looking at renting the car at an off airport location, your best bet is to contact the company/location specifically to see if they offer any customer pickup service.
Depending on the company, and your distance from them, they may be able to pick you up and bring you there to rent the car. If they don't have service for pickup available, check with them to see if they would consider reimbursement for your cab fare (if that would be your mode of transportation to get to them).
Airport car rental companies do not allow customer parking at the car rental parking lot either. But, if you wish to rent at an airport car rental facility, you can take advantage of airport parking (either long term or short term - depending on the length of your rental). The cost of airport parking will vary from airport to airport and you can find information about fees at any airport's website.