You know what everyone at my company uses for overhead door ops... Those large leaf spring center bolts. It's the perfect diameter for a nice fit and since it's threaded has a nice grip to prevent happy accidents. Also, it's the perfect length... Long enough for leverage, yet short enough to fit, (even if your an old timer who does them with the door up not down, or have two doors at 45degree angle on a box truck)
Some guys slide on a bit of heater hose to make soft grip handles. Best part, since it is neither made nor modified, just a long bolt, it meets tool box inspection standards.
Is there an official tool for winding up those door operators?
Also,lucky you being an outside vendor. For us, eeven vendors must be certified and follow our rules if they want to work on site for us.