I'm shocked that they would make you use your own car. I used to deliver parts for a few years, and still work at an auto parts store part time, but I'm behind the counter now. The store I work at always had the most random, ragtag bunch of vehicles you've ever seen for delivery vehicles. At one time or another, we had several Chevy S10s, a couple Chevy Astro vans, a few Dodge Caravans, a Dodge Neon, a Dodge Dakota, a Ford Ranger, a Ford Taurus wagon, and an Oldsmobile Cutlass coupe. Basically whatever the owner could get his hands on really cheap. They were often given to the store owner as payment by some of the shops we delivered to. They were all beat up, dirty, had close to or over 200k miles, half the accessories didn't work, etc. Just real pieces of crap.
You should use either a minivan, wagon, or small 4 cylinder pickup. If you use a pickup, make sure it doesn't have a slippery drop-in bedliner. Parts will slide all over the place. Not good when your cargo consists of 5 gallon buckets of oil and/or 100+ lb commercial batteries. A rubber bed mat would work well. A truck cap may also be a good idea. Most of the time you'll probably be hauling just a set of brake pads and an air filter, things that any car can easily haul. But sometimes a shop needs to restock their oil, or needs a large steering rack, or you'll be stopping at several shops on the same run, and they all have large orders. So something like a van or truck is required.