What's harsh to us, insofar as soap goes, isn't harsh to dogs. Their skin/hair pH is almost at the opposite spectrum of ours. Therefore what is drying to us, is moisturizing for them. What's good for us, is drying for them. That's why you're not supposed to use human shampoos (not even baby shampoo) on dogs. It'll dry their coat and skin.
However...the same conditioner can be used by both species.
I don't think you need glycerin added to the shampoo mixture. And, if I was going to use the vinegar (apple cider only), I'd use it as a final rinse. It's a great conditioner and makes the hair shine (for us or them).
Oh and what kills the fleas is soap, not vinegar (although it is used as a holistic repellent, and doesn't work all that well for any length of time). And start shampooing (with a puppy tear free shampoo) on the head then move down, because when you put the puppy in the water, the fleas will all run up to the head and face.
Lemon Joy has been a standard flea shampoo for eons. Don't know what the "lemon" does exactly, unless it helps to repel briefly. I've used it on baby pets for probably 40+ years (no vinegar or glycerin...but I do use apple cider vinegar and water as a rinse). However, it won't keep fleas off, so if the animal goes back into a flea infested environment, they will return within a day. That's why you need some sort of long term preventative. Talk to the Vet about which he/she recommends for your area.
If you decide on the mix and want to add glycerin, you can get it off the shelf right in a drug store (but I haven't seen it in WalMart)...especially a drug store that has a compounding pharmacy and all sorts of medical/home supplies. Make sure you get the USP Glycerin.