Yes, you need to progress in El Capitan's repeating mission "Just Another Story".

Assuming you have Phantom Liberty, vehicle contracts are repeatable missions that will only become available once you complete the third Phantom Liberty main quest called, “Spider and the Fly”.

I don't know how it works in the 2.0 base game, but the missions (indicated with a flashing steering wheel icon on your minimap) are popping up randomly in Night City too, not just Dogtown. And they are close to where you are, you won't have actively to search for them.

EDIT:
Note that the Quadra "Wingate" is currently bugged and cannot be obtained (is supposedly fixed in patch 2.01).

I'm in the base content still, super early (started over).
So I just have to advance stuff and should be able to get new rides then?

In the base game (I didn't start over on 2.0), from what I can find online (seems like the internet expects everyone to have Phantom Liberty now...), it is still tied to street cred.

Assuming the car system is similar to 1.6, it should still be tied into street cred. Does the autofixer terminals say anything about requirements or why the cars are locked? If you have Phantom Liberty installed, even if you have never visited Dogtown, I expect that to be the problem. But in that case, the terminals will tell you if you need to do more vehicle contracts.

And assuming you have played the game before, what is your current street cred? The best car (Caliburn, excluding the one you get for free through a mission) required 40 street cred back in 1.6, though 20 street cred for the Mizutani Shion "Coyote" was typically where I took a breather from the street cred grind.