Is it possible to move an Animal Crossing: New Leaf save onto a different game card?

I rent (and still have) Animal Crossing: New Leaf. Since I might buy it later, and I have gotten pretty far into the game, is there a way for me to keep the save so when I buy the game later I can use the same save?

It is possible in a couple of situations, but it doesn't sound like it is in your case.

The more normal way to move is to just move. IGN has a guide on how to do that here, but to summarize, you have to have your first house payment made, have another 3DS nearby with either a physical or digital copy of New Leaf already in it, and you have to not be your town's mayor, i.e. not the first save on your game. It sounds like you don't have another copy yet though or another 3DS, and you're probably your town's mayor, so that probably won't help your case.

Another option is to download the "Save Data Transfer Tool" from the Nintendo eShop. This requires you to have a digital copy of the game before you can transfer it, but it should work even if you're mayor.

If you just want to back it up for some possible case in the future though, I think you're out of luck. Some things do have "Save-Data Backup" as an option to the left of "Manual", but Animal Crossing: New Leaf isn't one of those things.