My Jonway 250 seems to have a stuck starter. When I connect the battery the starter engages and won't stop

solenoid incorrectly wired up

If the starter motor is turning the problem may be a broken flywheel tooth. If starter is not engaging the solenoid may be bad or not hooked up but that seems unlikely since starter was replaced. The problem's down there. If solenoid not clicking then problem may be further back.

YQou've pposted your Question in "mobility" scooters forum. The vehicle you are posting aboutt uses fuel. Generally we use electricity, Big Batteries. I lookd up your battery. http://www.monsterscooterparts.com/12-volt-ytx7l-bs-scooter-motorcycle-battery.html

Looks like a normal battery, it should be under the seat. Scooters makers somtimes put a storage compartment under the seat. So you must remove one of the side panels.