Hopefully you've made friends with the wrenches in the M32 ARV in your company. This the company maintenance section, the M32 is basically a tank hull, with the turret replaced with a crane system for lifting decks, engines, even entire turrets. If it is something which can be commonly and easily replaced, the chances are that they'll have a spare in the M32. Offer them a pack of cigarettes if they'll put you to the head of the queue.

In the event that the replacement is too time-consuming, the tank will be passed on to the Service Company of the tank battalion, which has a maintenance platoon and additional ARVs. (They don't want to keep the one maintenance vehicle in the line company busy for too long). Even a 6-ton wrecker will suffice for the hypothetical you pose, see example: https://s-media-cache-ak0.pinimg.com/736x/93/4b/3d/934b3d8045af44349de0b90cfa2be16c--ww-tanks-military-vehicles.jpg

The Battalion maintenance platoon will have a large number of reasonably commonly used spare parts, and probably a spare engine or two if necessary. If they don't have the spare part to hand (or the spare engine), they will likely send a request up the line for one to be hauled out to them by truck. Usually the divisional workshops would be something more serious than a mere engine changeout, it is unlikely that the tank would be dragged to a higher echelon.

In terms of actually having the spare part, it is one of the unsung war-winning advantages of the American system that all the parts were entirely standardised: Any part would fit any tank (As long as it's the right type of tank, of course). Especially in Europe, lessons about supplies were taken such that there was a concerted effort to push spare parts forward before they were needed. And when parts were in short supply, drastic measures could be taken. For example, DUKW (truck) parts were airlifted to Europe when it was discovered that they were wearing out far faster than anticipated.

If for some reason the US Army's supply system was failing, you could always trade with a neighboring unit. The last option is what we know today as "Controlled substitution". More commonly known as cannibalisation, take the needed part out of another inoperable vehicle.

If you have way more time than a normal person, and wish to learn all the US Army has to say about supply and maintenance in WW2, check out the history of Ordnance Branch in the field. http://www.history.army.mil/books/wwii/Beachhd_Btlefrnt/