Find a broken lawnmower in the trash for "bulk pickup" - free (getting it fixed might cost a bit of money, but in most cases, mowers get thrown away because the carb gets gunked up due to not being stored properly, or the air filter needs cleaning; ie cheap fixes).
Get the base or motors/wheels from an electric wheelchair (this might cost a bit of money, but should be doable for under $200 if you look carefully).
Various bits of steel, castors, bolts, etc - under $100.00 if you scrounge.
RasPi Zero W, GPS/IMU, sensors, etc (ie, electronics, sans motor controller) - $100.00, maybe a bit more.
Motor Controller - spend money on a good one, or money on the parts to build a good one. Probably looking at $200-500 dollars for a dual-motor controller. If you are really cheap, though, you can build a decent dual motor controller for the price of four 40 amp BOSCH relays, some wire, and one or two 40 amp DC SSRs; that'll get you a controller for well under $100.00, just don't expect great things from it.
The rest is up to you. Granted, that's the hard part, which may be worth the $2k depending on your skills, time, patience, etc.
Hopefully as they get older, we can combine spending time together and tinkering!
As an example, I'm in the middle of a stalled project now where I want to make nice looking fishmouth pipe joints. I decided to cut them with a hole saw. But I don't have an electric drill that's slow enough so I've struggled doing it by hand and with a windscreen wiper motor, all the time wondering if it would be worth buying a drill press, but not the cheap ones from the home handyman shop because their minimum RPM is too high. I saw a cheap used one with no motor online so maybe I could somehow get an old washing machine motor and build some adapters to fit it together and buy an electronic controller - and that would be a whole new project in itself! What a lot of fiddly, costly details to "just weld some pipes together".
Now draw the rest of the owl.