My veg patch was covered in Marestail when we moved in our house, 4 years later and it is still a slight problem but not as bad as it was. I too don't want to use chemicals and I pull it up using one if those deep root tools to try and get as much of the root as possible, although it isn't a field so that might be difficult for you
I started a thread on marestail a while ago when I didn't know what it was and there have been several others. See have said that the weed suppressant doesn't work, I don't know as I haven't used it. At the end of the summer I dig over the veg patch and take up as many roots as I can, I think this year I am also going to remove some if the existing soil and replace it with new topsoil/manure/compost mix.
To be honest I have learned to just live with it now and deal with it the best I can....I don't think there is an easy solution I'm afraid