As a former mechanic....
If I am correct, you wish the car to have run well a few days ago, and now will not start? Further, you want it to have add a bit of drama, and that a part will need to be ordered?
Why the requirement of a part that needs to be ordered? To keep it off the road for a few days? Weeks?
A good scenario might be a worn out timing chain. When overly worn (150,000-plus miles) or perhaps due to sitting (the chain rusted and causing it to get too loose), a timing chain on a car like that (probably a 289 ci) can jump a tooth on its sprockets. This frequently happens when trying to start the car, and so will fit your concept well.
It can add drama because when the chain jumps, the timing of the valves and ignition are off. This can cause the intake valves to be open when they should be closed and so a loud backfire through the carburetor can occur, usually followed by a bit of smoke.
Depending on where your story is set, it is possible that the local parts house will need to get the chain-set, and gaskets needed shipped in, and this could take a couple of days. You might also be able to make the shop too busy to get to the car right away, as the job will take many hours.
On an old Ford V8, there are other parts usually replaced at the same time, and these will include the water pump, fuel pump, and fan belt.
Hope that helps. If you have any other questions, do not hesitate to ask.
Shiny Side Up!
Bill