What does it take to get a feature request implemented?
First, you need to convince Stack Exchange that the feature is actually something that should be done.
Being popular is not enough - we are aware that Meta users are not, in fact, representative of all of the users and that sometimes, something that looks like a good idea at first blush is, in fact, a really bad one.
Now, some requests are all to do with the client side - things that user style sheets and user scripts can handle - and there are a bunch of those around (take a look at stackapps). These are the kinds of features that you can implement yourself.
As for other features - we have written and open sourced a bunch of libraries that we use in our codebase. If you want to work on features of those - they are listed on stackexchange.github.io.
The Stack Exchange codebase itself, outside of the OSS we publish, is closed source.
This means:
Is it possible for me to request to be allowed to do so?
Only if you get hired as a developer by Stack Overflow.