how common is it in st croix for people to go without a car. do most people have them, or are there many people who ride mopeds or do without?
It's not unheard of to go without a car (only know one person) but most people do have cars. A couple may only have one car instead of two here. There are really very few motorcycles of any kind on the island. STX is not bicycle or motorcycle friendly. The roads are mostly narrow and windy with many blind curves. Motorcycles require quite a bit of maintenance (not as much as a boat) so unless you're handy dont get one. Autoworld is now selling motorcycles but they only work on their own (ones they sell) or so I was told.
There are safari type taxi's (taxi's that function as a bus for a flat fee on a fixed route) and a bus that covers the island.
Once you find a job you like and plan to stay at, you try to find a place to live close to your work.
Yes many poor people who can't afford a car and/or illegals who can't get a drivers license have to get by without a car in the VI.
If the is the lifestyle you envision yourself living here then it is possible. You will end up spending your non working hours walking to do errands, shopping for groceries at nearby, but expensive convenience stores and you won't have access to many of the nicest beaches.
Walking after dark is dangerous as sidewalks outside of the town areas don't exist and most roads don't have shoulders so be careful!