The only way I know that works which is chemical free is to pull them all out by hand, hoe, then apply a barrier of some kind, either a thick enough mulch to stop more coming through, or a weed control fabric. Cardboard underneath either acts as a second barrier.
Vinegar will only kill seedlings, the easiest way I have found is to mulch the area at least 2inches deep at the start if the year and then remove the weeds that do come through by hand.
If you have got weeds like bindweed with rhizomes the only way is to clear the bed and use glyphosate because however careful you are you won’t remove every root fragment
I'm sure some lovely GN's will be able to help me out on this one. I am looking for a chemical and pet friendly way to remove weeds, roots and all, once and for all. I could look online but I bet someone on here has a magic formula. Thanks GN's.