I am a little confused on how this all works and help would be greatly appreciated.

Let's say I rented Little Big Planet Vita and I have beaten the first world. After doing that, I sent the game back and these are my two options:

1. Buy a used copy of it from a store.

2. Buy a new copy of it from a store.

Here are my questions about those two options:

1. Will the used game be able to use my save data on the Vita so I can continue from where I left off?

2. Will the new game be a able to use my save data on the Vita so I can continue from where I left off?

It should work the same as with any other console: the save file is independent of the game disc/cartridge. I suppose it's feasible that if you had played the NTSC version of the game and then imported a PAL version that the saves may not be compatible, but I see no issues otherwise. I haven't tested it so I can't say I'm 100% sure, but I'm fairly certain that's how it should work.