When a new feature is requested by the community and accepted with the intent to add it to the site, how is the addition actually implemented? Do moderators see the approved requested feature and pass it on to SE programmers, who then implement it, or do very high-reputation users actually have access to the site's source code and have the ability to edit it themselves?
how is the addition actually implemented?
Programmers. Stack Exchange employees that are programmers implement changes.
Do moderators see the approved requested feature and pass it on to SE programmers
I am guessing you are really asking here about how feature requests get to SE programmers.
Sometimes a programmer will see the feature request and will think good of it. If it makes sense they might just implement it. Sometimes they will think it should be discussed within the team - and after that discussion may implement it. All feature requests are reviewed by the community team. Some of these get passed through to the programmers.
do very high-reputation users actually have access to the site's source code and have the ability to edit it themselves?
No, they don't. Access to the source code is limited to Stack Exchange employees (except for all the stuff we open sourced).
There are a few issues with your terminology...
First off, SE is run by staff. Some of that staff writes code, some of it designs the look of the site, some of it is dedicated to the community relations (CMs), building sites, administration, HR... etc... lots of staff. These people were hired to work for the SE network and are paid to do so.
Moderators are not staff. Every site has a group of diamond moderators who are volunteers (unpaid) and help deal with the site's content and the users on the site. They have no access to the code but do have additional access to some user info and site metrics.
High rep users are high rep users. That's it. They get fun tools at different levels of achievement but they don't get any other special access. A moderator with 2K rep has more access to the site than a super user with 200K rep.
Now that that's clear, the only people who can implement new features, are staff.
But! Everyone can help make SE a better place by suggesting new features and you're in the right place to do that. Any user can create a feature-request question on Meta Stack Exchange, which is then voted on by the community. If the FR gets sufficient support, it may be considered by the site for implementation... even with sufficient support, the staff may decide that the request is either a bad idea for the site or too complicated to implement (or any number of other things).
So, while we can suggest features, we should not expect them to be implemented, though we can certainly hope that they will be and, if not, hope that one of the kind staff members will take the time to explain why the feature request is status-declined.