have you really thought about this?
I'm sorry if I sound harsh here, but I really question if you have given anywhere near enough thought to the plans you are proposing.
A car does not make a great bed, but it can work for some solo travelers looking for a cheap place to get off the road. As Lifey mentioned, Sleeping in a car is a challenge for 2 people, even if you get a larger vehicle like a van.
The fact that you'd even consider getting a compact car that you'd both sleep in, to me is a big red flag. The issue isn't privacy, its comfort, the ability to get your needed rest, and to actually be able to have any fun!
On top of that, the fact that your co-traveler is your teenage daughter complicates matters even more. Personal space is always needed on a roadtrip, but when teenager children are involved, triple it! I frankly, can't imagine any trip with a teenage child where sleeping in any car is an option, at least not one where you'd finish the trip with good memories.
I'm not trying to kill your dream of getting on the road, but I'd really think long and hard about your plans, and what you will actually get out of it. I have a feeling, you would be much better off if you waited a little while longer to save up some additional funds, or until you can take this trip yourself.
And now for the other side of coin....
If we have waited until we had sufficient money to create a business, RoadTrip America would never have been created. The original co-founder and I decided to launch RTA and all that comes from it with no discernible financial support. The same is true with a number of road trips I've personally embarked upon. For me, the risk of running completely out of money, was part of the appeal... I'm less comfortable with the concept these days, but my basic approach to business decisions hasn't changed all that much. Now, here's a book review about by Jeff Blumenfeld's book "You Want To Go Where?: How to Get Someone to Pay for the Trip of Your Dreams," that you might find interesting. It's not entirely a perfect match for your plans, -- but if you read it -- I think you'll know why I recommended it to you.
The RoadTrip Advisors and Enthusiasts who have responded to your questions have all raised valid and important issues -- and I hope you're able to use their hard-earned advice and expertise -- But there is also something to say for just doing it anyway and see what happens....
Mark