Take a few days to get to know the code, if you can afford it. Take a copy of the code base that you will absolutely massacre and get it up and running. Now do the following:
Most modern code on webapps (and Desktop apps, too) have almost all code initiated from the user's perspective. So take any user function and find the code that corresponds to it (this may be trivial, which is great). For instance, if the button says "login," change it to say "not login" or whatever. Now you know that you're editing code that is related to the app. Now start going forward in the call stack to get an idea of how the app is structured, what the layers are, etc. When you need to make sure you're right, break something completely or change it a bit to make sure you're in the right place. You can always get a fresh copy or just leave it broken, since it's a copy.
After you've done this with a few pieces of the app you'll get the idea that it's (hopefully) structured the same way throughout all functions. For instance, "add a client" looks like "login" etc. Break stuff and make sure you're right. Then put it back (or not) and keep investigating.
As you investigate, take chronological notes (I call this "rambling"). When you finish enough investigations to feel like you know what the app looks like and how it works (you usually get bored after you've finished less than 5% of it, since you can figure out the rest as you go), takes your "rambles" and convert them into a Wiki (as you said).
When you've finished this initial investigation, you'll be ready to hack into the app and make changes as needed. Not that you'll know the entire thing, but you'll feel comfortable enough to cut in, investigate some more, and make changes.
Note: if the earlier dev didn't have a version control system setup, you must do that and all similar infrastructure tasks. But, do stuff as you have time, and little by little, as always.