Is the speed of causality $c$ or $\frac{c}{\epsilon}$ where $c$ is the speed of light in vacuum and $\epsilon$ is the dielectric constant of the medium?
I searched the net but could not find a good and relevant answer. There was this but it did not address the main concern.
So, will an electron travelling at $0.99c$ in water, where the speed of light is actually $\frac{c}{1.33}$, violate causality?
It's always c. But in dielectrics the photons interact (scattering etc) with the media and sort of zigzag before going out (that's a simplistic way of saying all those interactions take extra time). So it seems they go slower. Anything that has fewer interactions will go faster, eg neutrinos.
Nothing will go faster than c. There is no causality issue.
To add to Bob Bee's Answer: to understand this situation, it helps to divorce light from special relativity and take an Ignatowskian approach to the latter, as I discuss further here. There is a universal signal speed limit $c$, and, experimentally light is found to move at that speed, or at least the two speeds are mighty near one another. That latter, experimental, result can be taken as asserting that light is mediated by something with zero rest mass.
So now, anything else to do with light - speeds in media and so forth - is simply not relevant to the causality issue, which arises wholly from the relativity from symmetry alone standpoint conceived by Ignatowski. The issue with $c$ is that relative speeds greater than $c$ can reverse the time-ordering of events for different observers, which is a real problem for a notion of causality if the two events in question are causally related: such as my switching on my gas ring and sometime later eating the boiled eggs that I have cooked. So, we simply postulate no faster than $c$ travel to keep our notions of causality in accordance with what we observe.
