Are you sure it's ragweed and not mugwort? If it's something that grows from runners, like mugwort , thistle, or bindweed, the only thing that I found to work was to dig up the whole area until you have all or almost all of the roots up, then be vigilant for the next couple of years and pull each tiny part that shows up. Don't till, whatever you do, as each little piece of root will form a new plant.

There is no "weed" killer that can be used safely, whether in your vegetable garden or elsewhere. Any plant poison you use will also kill your vegetables.

Ragweed, Common or Great, is an annual and in my experience it does not have a real extensive root system, although in some soils pulling it can be a bit difficult if the plant Is two feet tall or more. Around here cutting Ragweed off at soil level is often enough to stop it..

Bermuda is an invasive grass that you need to remove every bit of root from the soil if you do not want it growing somewhere and place good barriers to prevent the roots from moving back into that area. Newspaper and cardboard, or any other mulch material, will not work controlling that or any other invasive plant.

kimmq is kimmsr

I agree with both above that it doesn't sound like rag weed. They aren't on runners like that. They should still be relatively small at this point. At least they are in my zone five. They are extremely easy to pull at this stage and even when they get to full growth I can easily get them right out of the ground with my hands if the soil is slightly damp. I live in the country and I get a lot of them. Straw mulch does keep them down somewhat and just keep pulling.

If you post a picture, here or on the Name That Plant forum, we can get you an id. That's the first step.