Strip the go-kart down to the chassis. Sandblast down to the metal and inspect the condition of its frame. If any cracks or breaks are apparent, some spot welding will be in order. Use lubricant to loosen any frozen or sticky parts like spindles, tire rods, or steering wheel until they are moving freely.
Straighten the axles and weld the go-kart floor pan if needed. Prime and repaint the body. It may take several coats of spray paint to get the desired appearance. Wait up to a day for the paint to dry completely.
Source out replacement parts based on your needs and the model of your go-kart. Period parts can be located on the Internet, allowing the go-kart to keep its vintage look. Consider contemporary alternatives as well.
Replace the tires on the wheel rims. Some models of tires will have their make and size stamped on the sidewall, so locating an exact replacement may be possible, but new tires with fresh tread will allow for a smoother ride. The wheel rim diameter will be the key to locating tires of appropriate size. Full wheel replacement is desirable if original wheel units are damaged beyond repair.
Install the motor. If motor is missing or inoperable, the appropriate replacement for a gas driven go-kart with a 200-pound body maximum will be a 6.5 horsepower motor. The cost of buying a new motor is reasonable, and may be a lot less trouble than getting a disabled one running again. Fit, but don't force, the clutch and governor onto the output shaft accordingly.
Bolt on seat or replacement seat. Reattach brakes. If old breaks are worn out, mini-bike breaks make a great replacement. Adjust the throttle linkage and torque as needed. If the go-kart does not have a kill switch at the steering wheel, install one for safety reasons. At this time the go-kart should be operational and ready for riding.