I wouldn't use herbicides or harsh chemicals in your vegetable garden. Some of them linger in the soil so while you may get rid of the grass, you won't be able to grow vegetables either. Part of the answer depends upon what type of grass you have. If it is a clumping grass, manually pulling and mulching will keep most of it from growing. Do know that straw had LOTS of seeds. I use straw as mulch sometimes, knowing that I will be weeding a lot in the first few weeks.
If you are dealing with bermuda grass, you may want to consider a raised bed. Each time you till bermuda, you break up the rhizomes and create even more plants. It is awful, I am currently digging up a flower bed where I lost the battle