You don't need to craft to play the story, BUT, there are a few story quests that require a crafted item that they tell you to craft.. Just go to the bulletin boards to get it.

Also, fyi, all crafting classes can be leveled to 20 WITHOUT gathering. The mats can be found on vendors for next to nothing. Some are in the crafting vendor, some are in city vendors, some are in outpost vendors, but up to 20, all crafting mats can be bought from a vendor, after level 20, some still can be bought from a vendor some need to be gathered or farmed off mobs.

The only thing that will hold you back from doing the story are the mandatory dungeons and waiting in que for xx amount of time. Everything else can be done.