Not a darn thing wrong with trying to keep it ( cheap ) inexpensive. That is a fundamental part of a successful life in my book. I am not very schooled on types of welders. I believe that the flux core type are a lot easier to use than the
arc welder, but you probably should get others opinion and or do some research. If you do build it from scratch there are plans available for many different styles and sizes. Also before you begin or buy a welder you should figure out material type ( square or round tubing ) wall thickness, total amount of lineal feet needed ( plus 10 to 15% ) for mistakes. Find a metal yard and price it out. Then make a list of incidental parts you will still need to purchase.
Rear axle and bearings, bearing hangers, motor mount plate, front spindles, tie rods, brake system ( mechanical cheaper, hydrolic more but works better )
steering shaft and wheel, seat, cables for gas and brake. On top of that you still have the welder to purchase AND don't forget a power chop saw. NO WAY
are you going to be able to make the amount of PRECISION cuts necessary with a hand hacksaw and not end up with a lopsided mess. I am not trying to discourage you, I just want to make sure you have thought and PRICED this project all the way through. Those $200 frames might start looking better. If you do build one, with 5 kids, I would recommend it be a two seater. Maybe you should build two. It would help justify the tool investment you are going to make. One more recommendation, bolting a steel frame together would not be much more riged than wood after a very short time.