Answer: I would not recommend using a chemical weed killer in your vegetable garden. While all gardeners will have to do some weeding, there are steps you can take to minimize weeding.
Here are some tricks I use. Cover the entire garden with black plasticor even tarps for a few weeks in early spring. This helps kill early sprouting weed seeds. When you remove the plastic, till the soil one time. Wait a week or so, then till again.
Mulch the pathways heavily, so at least you won't have to weedthem. I use a thick (a solid one inch) layer of newspapers laid flat in the row, followed by a layer of straw mulch. You could also use black plastic, landscape fabric or a very thick layer of grass clippings or straw.
Plant seed in rows, so you can easily hoe between the rows (rather than scattering seeds). As soon as seedlings are up, begin mulching around them--I use grass clippings piled thickly between plants.
You could even put a layer of landscape fabric down, cut holes in it, and plant your seedlings in the holes. This is especially good for tomatoes and peppers--plants that need to be widely spaced. You may want to add a thick layer of straw mulch around each plant, so the soil doesn't heat up too much under the dark fabric.
To keep the surrounding lawn from encroaching, I dig a deep (6-8") trench around my entire garden. Then I make a weed-free border using a thick layer of newspaper followed by grass clippings or hay. I can run the mower right up to the border, andany weeds that do try to climb over the border are easily removed.
There is a new natural herbicide available, made from corn gluten. It prevents seed germination. I plan to give it a try this year. Just remember that you will only be able to use it around established seedlings--don't spread the herbicide, then plant seeds or it will inhibit their growth too. (WeedzStop is available from Gardener's Supply, http://www.gardeners.com, ph# 800-863-1700
All these things take a little time in the spring, but they are well worth the effort, and will save lots of time later on in the season. Also, each year you will decrease the weediness of the garden.
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