Vinyl is generally a fairly "soft" plastic with high damping for acoustic frequencies. Making a mandolin out of it would be about like making one out of rubber. Sustain would be expected to be somewhere between "lousy" and "you've gotta be kidding," although if you worked really hard at it you might make it sound like a banjo.

In some grades a polycarbonate or other "more exotic plastic" might be a little closer to the acoustic properties of more traditional materials.

Since a violin (or a hurdy gurdy) has a sustained input from the bow (wheel) sustain is less of a problem than with a plucked/struck instrument, but tone quality would still be something of an open question.

In either case, most of the more common plastics have a tendency to "creep" under load. They may be strong enough to hold the load without breaking; but will change their dimensions if the load is there for a long time. Even with wood, only some woods have appropriate creep resistance to be useful for the parts that hold the string tension.

The few stringed instruments successfully made of "plastic" generally have used very high "fill ratios" - with lots of carbon or glass - to "harden" the material and reduce the creep tendency, and use "plastics" that you're unlikely to find (affordably) available without significant investment in processing machinery and handling equipment.

You might get yourself a good TIG welder and try alumin(i)um if you're just looking for an adventure.

John