Many, though not all, off-airport locations (often franchises, I think) of various rental car companies (not just Avis) will pick you up within a certain range (e.g., 5 miles), but normally NOT from an airport (violates the company's agreement with the airport rental facility). The rental conmpany vans that regularly go by the terminals are for the airport rental facilities. So if you want to use that off-airport rental facility, you'll likely need to take a taxi or at least first go to an off-airport location. Call the rental facility to confirm their rules. Google Maps shows the distance from the airport to the rental place as just under 3 miles.
http://la.taxiwiz.com/?lang=en estimates the taxi fare at $12.86, including the airport surcharge. I doubt you'd fit a family and 11 pieces of luggage in a regular taxi, so you may have to use one of the larger vehicles (e.g., station wagon) or use two taxis or you could go get the car and then pick up your family and bags back at the airport.
I've used off-airport rentals at times; sometimes they save you money and sometimes they don't. Some have limitations on one-way rentals and even locations to which you can go (in terms of states). And once I had an off-airport franchise of Budget insist I go to a car wash or pay them $25 to clean the car since if the car were dirty it violated the terms of bringing the car back in its original condition (pure b.s., but I didn't have time to dispute it).
Off-airport rental facilities are normally much smaller than those at the airport, and will have a smaller selection of vehicles, but if you can live with that, they can work out well. You are, of course, likely not guaranteed a particular make or model of vehicle (usually says "or equivalent"), so you may or may not get a Caravan, but that also applies to on-airport locations.
Just for future reference: I've used discount codes from AARP and several airline frequent flyer programs (never been asked to produce proof of membership in a ff program, but was once asked to show my AARP card). One friend said he got a super rate for a San Jose-San Diego rental by using Southwest Airlines's car rental feature. And both Priceline and Hotwire have car rental features. On Priceline you can even bid.