In the face of it, converting a sewing machine shouldn't be all that difficult - hence the idea. You have a needle that goes up and down at a speed fast enough to drive the saw blade, a work area with a section underneath to keep tension on the blade... What could be simpler? All you need are some basic hand tools, some epoxy, and an old unused sewing machine.
Step 1: Assess your sewing machine
There are lots of leavers, pulleys, gears and pivots. Even if some of these have fallen into disrepair (as had my machine) this typically will not matter, as we will be stripping lots of it out. Compared to a sewing machine like this, the scroll saw I have in mind is actually quite simple. It helps if the motor is working and the needle is going up and down - everything else is pretty much surplus.
So unscrew the case retaining bolts and have a good look and what you have got... It might well be the case that you will have to modify thing to suit your particular machine.