You can certainly level up just the gatherer classes on their own.

Of all the crafter classes Culinarian is the one with the least dependencies on other classes.
But even so, if you are going to do any crafting I'd strongly recommend leveling all the crafter classes to level 15 at least for the first set of cross-crafting skills. For high-end crafting you will ideally want to level up all the crafters, but just getting them all to level 15 will get you quite far.
Getting to level 15 won't take more than an hour or two, and all the materials needed for the level 1-15 recipes can be obtained at the crafter guild or elsewhere in the same city.