General aviation pilot reported a traffic conflict on takeoff; when a vehicle crossed the takeoff runway without an ATC clearance. The pilot avoided a collision by continuing the takeoff and climbed above the vehicle over the runway.

Narrative

CFI candidate training flight with a part 61 flight school with another commercial pilot on board acting as the student. RTB (Return to Base) approx XA:15 to practice teaching patterns. Around XA:30; we started a practice short field landing from the downwind; landing was being demonstrated from the right right seat. Tower gave clearance to land runway 32 on the downwind abeam the 1000 ft markings. Initiated the descent and configuration IAW (In Accordance With) standard procedures. While turning final; tower instructed a truck to hold short of runway 14/32 at the approach end of 14 where taxiway F and N cross runway 14. Over the radio the fuel truck acknowledged instructions. Practice landing touched down on the 1000 ft marker; and pilot flying began touch and go procedures. At approx 55 indicated airspeed on the takeoff roll prior to rotation; the fuel truck began crossing at the approach end of runway 14 directly in front of the departing aircraft. ATC did not issue runway crossing clearance for the fuel truck. Due to the higher energy state of the aircraft being so close to rotation speed; I elected to commit to the takeoff and rotate. We crossed over fuel truck around 100-150 while it was still on the runway. Tower acknowledged the violation over the radio.