Long story short: I'm in Wellspring and I can't play the Buggy/Rat Rod races that require a rocket launcher, because the vendor is not offering a rocket launcher for these cars. The only rocket races I'm eligible to do involve the Cuprino (as it has both, miniguns and rocket launcher from start).
I was wondering if I need to find the "on-foot-"rocket launcher and this would be then used for the other cars as well - but a friend told me that he simply bought a car rocket launcher from the vendor. He cannot remember anything he would have done to unlock them, though...

Does the car RL show up when I progress further? I'm about to venture into the dead city to get to the hospital.

Seam to recall that that you have to earn the guns and rocket launcher from a mission (from the sheriff).

The handheld rocket launcher is given to you in the dead city, It has nothing to do with the cars rockets, they are two separate items. The handheld can also be found in one of the sewers (before Dead City mission)if you have the anarchy edition.

Also you have to 'TUNE' (chosse) the rockets or the guns on your Dust Buster in the garage. You can't switch on fly.

I came back from the dead city and the car rocket launcher was offered to me at the vendor's.

So I guess you're just not meant to buy it before advancing the main plot to that degree. I can't think of any other thing I would have done to make it show up.