Crawlers are the infant Necromorphs that first appear in Chapter 6.
To kill them without detonating them, you can use the Plasma Cutter and shoot off their head. This will kill them in one shot, though can be tricky since they will be crawling towards you, and if your aim isn't true, they will explode and possibly harm you.
Alternatively, to make this trophy trivial. You can use the Force Gun (found later). One shot will kill them, and they will NEVER pop (Unless you use it's alt fire. Don't.) Depending on how well you do, you will get this trophy your first playthrough. Or, you may need to obtain it in NG+.