This is very unlikely to ever evolve, for an animal to have two forms there needs to be a reason for it to have both forms. Each needs to provide an advantage that compensates for the costs of changing. In the case of a werewolf the wolf form is clearly a superior hunter - but what would it gain by being in a non-wolf form for most of the time?
Also consider animals like caterpillars and tadpoles that undergo transformations in real life and look at how long those transformations take. It's not a simple matter to reshape your body.
Spreading by bite
This is clearly possible if it's a virus of some sort (or even nano machines) being transmitted in saliva.
Triggered by full moon
This is hard to explain, it would be easier to link it to some other cycle and have it coincide with the full moon. The one advantage of the full moon is that the nights are brighter so you could make some argument for it needing that but it's a pretty weak one.
The actual transformation
If mass changes then it's basically impossible by physics as we know it. That means you're going to have a really big wolf or really small person. Take a look at this question for a detailed discussion: Is there a credible way a shapeshifter could gain/lose body mass when changing forms?
So let's say a human turns into a huge wolf. In order to do that they need to reshape bones, re-align muscles, stretch skin, grow fur. That's not changes that can be done overnight. Any realistic biological system is going to need months.
If you had special transforming bones that could possibly speed that process up but would weaken the bones drastically in the process.
So in other words, no it's not possible.
The closest you could get is something which made more superficial changes. You would stay humanoid but you would sprout hair rapidly, claws normally sheathed would extend from the fingers, the jaw would hinge open wider and protrude, the teeth would also extend.
These sorts of changes could theoretically be done much faster, in minutes for the basic changes and in hours to grow the hair. In fact if the hair was always there but retracted inside the skin then that too could grow out very fast.
