I can't find my skewer that came with the trainer. Can I use my bike skewer? I have misplaced my trainer bike skewer. Can I use the one from my bike?  I am not concerned about scratches.
I wouldn't. You'll damage the skewer you have, which will make it unreliable on the road, if not unusable.

Basic skewers with plastic on the nut will certainly be damaged, the plastic gets crushed and then you can't use it. Even skewers with more expensive metal parts will be subject to forces they weren't designed for. If a nice skewer should be damaged in this way, that is more expensive than the right trainer specific part, it is a false economy.

I should have your LBS order in something suitable