you had to choose between vinegar and glyphosate, I'd use the vinegar. your dogs aren't going to roll around in a too-strong concentration of vinegar, it has a smell and if it were actually going to cause burns (what kind of vinegar is this, something from a chemical supply house? ) animals would be immediately repelled by it (plus it evaporates quickly anyway). whereas with glyphosate, none of that applies, it's a fully synthetic chemical that stays in the atmosphere for days, would not send any cues to animals, and its effects on animals may be long term, concealed for years, and fatal.

but as someone else said above, if this is a certain area that your dog wants to be, you can always pull weeds for that area by hand, just make sure you get the entire root.