No, they cant use it, so they KILL little sisters, SAVING requires an specific Plasmid. They cant salvage it trought bodies, mainly cause of the missing ADAM recycling part from stomach, ONLY little sisters and big sisters have this required part of the ADAM making circle.
^ Because other people don't have the slug in their stomach doesn't mean there still isn't ADAM in the body. That recycles ADAM and creates more, which is what the Little Sister does. I'm guessing it is possible to get small amounts of ADAM from corpses, similar to BioShock 2 when adopting a Little Sister. If Little Sisters are able to get ADAM from a corpse by using a needle, then so can Splicers.
It was never really talked about in-game for gameplay reasons, although there is nothing suggesting it isn't possible universe-wise. The ADAM taken from a corpse probably wouldn't be enough to do anything with, so they would have to use a needle on multiple bodies.