How many free cars are there, and where do I get them?

You can get a free Mizutani Shion “Coyote” if, during the quest With a Little Help From My Friends, you inform Saul about Panam’s plans, rather than helping her and the crew out.

This will lock you out of Panam’s remaining quests however.

There is also a Colby CX 410 Butte just sitting, waiting to be claimed, at an old abandoned film set in the Badlands, not far from the Rocky Ridge fast travel point.

Finally, players who started with the Nomad life path will be able to reclaim their Thorton Galena from the badlands junkyards. You’ll receive a message in game starting a quest to go pick it up, and either pay 100 Eddie’s, or simply threaten violence, to reclaim your ride.

It's perhaps worth pointing out that you can also get free cars at almost any point by carjacking a car you see nearby, whether it's currently being occupied by someone or not. Some cars are locked, but many allow you to oust the driver or simply get in the driver's seat and use the car (including police or MaxTac vehicles) without raising your "wanted" level. If the car has a driver in it, you may need a certain strength level (e.g. 8) to pull the driver out. The necessary level will be shown next to the dialogue option, just like encountering a door you can force open.

Such cars aren't "owned" by you and can't be called to you after taking one and then getting out of it, and you can't use them to access your Stash, but they are technically free cars.