Ouch, keep an eye on that big chunk, does look like it's the fin and didn't get into the body. Does look like a fish, by the look I'd lean more towards the puffer. They have that v shaped beak, thing is the dogface wouldn't have to be aggressive more like looking for food. The wound at the end of the tail can also be troublesome, it can start to spread up. These things have a tendency to move fast and when they do many times get out of control very fast. While I'm not quick to jump on antibiotics, I would be inclined to seriously think about it. Is the big chunk and the tail wound on the same eel. They would need to be removed to a hospital tank to treat with antibiotics.
@Jay Hemdal what do you think
IMO - the first thing is to remove all "nippy" fish from the tank, the puffer and the trigger for sure, but also cleaner wrasse, etc. I've never seen that sort of aggression being well controlled (it is probably food/curiosity motivated). I've seen butterflyfish pick at wounds like this, apparently just being curious. Have you noticed this? Ribbon eels don't seem to have a good "sense of presence". If a fish swims up to their face, they back away. However, if a link of their body is exposed, a fish can swim right up and bite it and the eel doesn't seem to try to avoid that, almost like they don't know where their body is half the time.
Here is the issue with antibiotics - if the OP can't set up a stable treatment tank, I'm not sure I would risk it. Ribbon eels don't take well to sub-par aquariums, both in terms of hiding spaces and water quality. I'm afraid if they move the fish to a marginal tank, the fish will just go downhill. If they do try antibiotics, a broad spectrum, gram negative one would be the first choice - Neoplex, Furan-2, Kanaplex maybe.
What is that disseminated mottled coloration on the one eel? That worries me more than the bites do.
Sorry, I don't have a lot of recent experience with ribbon eels. Through the 1970's to the mid-80's I had a 100% record of
none of them accepting food, I tried dozens of them. Then, I got a juvenile in 1984 that ate. That was so unusual for the time, I wrote an article about it for FAMA. However, like orange spot filefish, I just stopped buying them, too much work, and too high of a mortality rate. Then, over the ensuing 30 years of so, something changed - and while still delicate, more people are having some success with both species. I suspect that it was cyanide issues in the past that have become less common.
Jay