Touch is not defined in 5e
Even "unarmed strike" isn't very well defined. Part of the simplification that went into 5e means that we no longer have stuff like 3.5's "touch AC", which means it is pretty much up to the DM. However, we can get a bit more insight via the Sage Advice Compendium by Jeremy Crawford, because in it he says several times that
unarmed strikes count as melee weapon attacks despite not being weapons
From that sentence we can infer that an unarmed strike touches the target and deals damage. It then follows that if that touch were to be a Grung's skin, it would poison the attacker.
The bigger question that we are left with is when exactly does the attack touch the skin. That I believe is again left up to the DM, with attention to the circumstance at hand. One could do this methodically, but then a lot of questions arise very fast. If an attack passes a creature who has no armour's AC, then it clearly touched them, but then does an unarmed strike touch a barbarian who has Unarmoured Defence even if it didn't pass the AC? How about a monk? If the target wears armour, does the attacker need to specifically try and hit the areas where the skin isn't covered, or does it just need to be extra good?
These questions are answerable with some thinking, but those answers will likely never be universally agreed upon. And even if you come up with the answers, your PCs will come up with a situation you definitely didn't think about, because that's what PCs do. So my advice is to just try and judge the situation, and improvise. Maybe you could make them roll a Dexterity saving throw in addition. Maybe you could lower the DC on the built-in save. The world is your oyster, and whatever solution you come up with isn't too important so long as the game is fun and immersive.
As for the last question, it seems like you could make them roll for every attack. However, if you don't want to kill your monk, you could consider adding caveats à la Frightful Presence, where a creature who succeeded the throw is immune for a set period of time. Or maybe don't! Forcing your PC's to be creative could be fun as well!
TL;DR: Unarmed strikes probably count as touch.
The rest (does one need to roll a save for every strike, how armour affects the situation etc.) should be up to the DM, with attention to the circumstances.