How much would it cost to reinstall ductwork under my house? Cats have shredded over half of it.?

1st go and get a trap and catch the cats to get rid of them. 2nd with out knowing how many runs of duct. Guessing 1000-1500. But to be honest if you know someone handy you can buy the flex duct at lowes, or menards. Its not that expensive and is held in place by zip ties.

So your duct work was not sheet metal, but some other flimsy material that cats can shred? If you have a crawl space vs a basement its probably going to cost extra because of the difficulty in working in a crawl space once the decking is on the floor. If its an unfinished basement, you should be able to get some kind of rough estimate over the phone from an HVAC company

Sounds a lot like a manufactured home. If it is, all you have to do is call any manufactured home service company in the phone book and give them the size of the house. They'll tell you over the phone.

Have them fix the skirting, too, or you'll be doing it again soon.

Probably about $5000. But if it was able to be shredded by cats, then it couldn't possibly have been metal to begin with. It was sub-standard. Consider this a blessing in disguise. Now you will have proper ductwork installed.