Chemical weed killer => ground => dog paws => dogs lick => into dog digestive system
Perhaps not what you want the dogs to eat?
For tap-rooted weeds, I like to use a tool I call a "weed stick." It doesn't seem to have any specific name, but others have told me it's a dandelion weeder, an asparagus knife, and a few other oddball names. It's basically a rod approx. 12" long beyond the grip with a slightly bifurcated end. You put it into the ground as deep as you need to, wiggle it around, and it usually pulls the taproot free enough for you to pull up the whole thing.
Otherwise, boiling water will kill weeds and grasses. Unfortunately, if you apply it to earth vs. pavement, it will also boil earthworms....
Kitchen vinegar, undiluted, can also kill plants, as can horticultural strength vinegar. I haven't used anything beyond the weed stick and boiling water, so I don't know where to purchase it, but a phone call or visit to an independent garden-supply store should produce all kinds of information!
Sunset Zone 17, USDA Zone 9