@Rclear I am the race director at PittRace. What you are calling a street track is called sprint kart track. No idea why they call them that, I haven't been around the sport long enough to know. My kid has also raced dirt karts including at Goodhope near you in Columbiana, OH. What you want to get is a kid kart chassis with a Comer 50cc motor. Popular brands are top kart & Birel. Anything from say 2003 and up is fine, no need to waste money when starting. You can actually race that chassis/motor combo on both asphalt (sprint) and dirt, so with a simple change of tires you can run both.
In sprint karting you have the following chassis sizes
kid kart
cadet
Sr
In dirt karting you have
kid kart
Sr
Trust me from experience, fitting a 5 yo in an adult sized go kart is a PITA and not worth the effort. No idea why dirt karting doesn't make a mid-sized chassis, but they don't, all they do is some Jr drivers run a thinner walled chassis, but the fitment issues remain.
Here is a photo of what a older top kart chassis would look like
https://trendsetterspowersports.com/listing/top-kart-kid-kart-comer-50/
You can also put a "dirt" or "gold cup" style body on them, and in fact Mark Bergfelt, who lives down the road from you sells the best one in the country. We allow that style on the sprint track as well, so again less changes if you wanted to try both.
For motors, you have WKA rules and Outlaw (just about anything goes). Mark is the best builder in the country for outlaw and he also does WKA ruleset comer motors. By rule, we don't allow outlaw motors at Pitt. We follow WKA rules, which long story short is what most dirt tracks claim to run as well, but I've never seen a comer teched at a dirt track EVER. Basically an outlaw motor would out-perform the same motor built to WKA rules, but the driver has to be really good for it to matter. IE, if you found a smoking deal on a kart and it came with an outlaw motor, unless your driver was Mario Andretti, I wouldn't know the difference unless I tore it apart, which we don't do to comers at Pitt (it's a kids class). Hope that makes sense, but tl;dr, you shouldn't cheat, but I wouldn't know unless I paid Mark to come tear it down or your kid got so good they lapped the field.
Feel free to PM me, I can probably help arrange a test session for you, etc. I think opening day at the track is April 7th, our first race I know is May 8th.
As for tools, trailer, etc. We had a kid kart family last year who strapped the kart on the roof rack of his Jeep every week. They had a small tool box and that was it. There is nothing you need that someone else at the track doesn't have when you are starting out. Just ask. Don't go all broke buying chain breakers and tire bead breakers who you can just ask and borrow. We have been doing this 5 years now and we still pull our stuff on a flat bed trailer with a little Mazda 3 and we've gone to Indiana the NC to race, including this weekend.