I've been working on these and I've found it annoyingly inconsistent.
As others have pointed out, sometimes you can take a vehicle from your garage. There doesn't appear to be any consistency about when this is possible and when it is not. I have found that for missions where there is not a requirement to go to a certain district, it seems to take a vehicle from your garage more often.
Most of the time though, you have to travel to the marked district and steal one of the cars that show up on your radar with a white "car" outline. I've found cars before the point at which I entered the marked district and driven them into the district before (both ones I already had in my garage and others that I'd just stolen along the way) and the game sometimes doesn't count these cars. You can get out and back in, and it still wants you to drive around until one pops up on the radar.
In some cases, there's a very specific car you must steal - one of the later Humanitarian Aid missions, for example, wants you to steal a particular Falcon VTOL and will only accept this VTOL, even though there are 4 parked in a row in the area they send you to. In many cases if you've entered the area in a flying vehicle you'll immediately get a "the target vehicle was destroyed!" message, even if you're over top the target vehicle and can see it clearly from the air.
One thing that I've found that works for most cars is the 'police notoriety wipe' function in your phone. If you've unlocked this, you can generally notoriety wipe yourself once you're in the car and proceed to the drop off point relatively unmolested by the police - just stay in the car. The police will tend to fire on you and frequently destroy the car 90% of the way to the chop shop, so they're sort of the biggest threat in some of these missions.
The other important upgrade is the "nitrous everywhere" upgrade, which can make some of the larger vehicles slightly less annoying to drive. Just watch out, as sometimes hitting another car in a large vehicle while firing nitrous causes it to explode, or causes you to eject yourself from the car, thereby causing you to get your notoriety back.
The one major exception to this is the tank - driving the tank increases your notoriety steadily as you go, so make sure you have both notoriety wipe options if you can. (one is a story unlock that you can opt not to take, and the other is an upgrade in the phone)
Also note that especially when taking cars to the airport, the GPS lies and takes you on the worst route possible. Check your map if you're driving to the airport - this is especially important during the tank theft.