When describing how a site is built, there's usually two terms: static vs. dynamic.
Typically a static site is just HTML, CSS, and JS files that are uploaded to a server. To update the site, new static files get uploaded manually.
A dynamic site uses some form of back end technology to automate the updating and maintenance in someway--often including the addition of a Content Management System.
AFAIK, there is no particular word that would be the 'verb' form of 'dynamic'. I think for a proper term, you'd simply say "to make dynamic".
As for the other suggestions (Energize, Animate, and Interactivate) I'd say these are all words that have no bearing on the back end. These would all describe front end qualities (like interaction and animation).
@sgroves's "dynamize" is the perhaps the closest to an official way to turn 'dynamic' into a verb. Your 'dynamicify' is not a real world, but hey, it sounds catchy. Maybe go for it.
Personally, I simply say "add a Content Management System". I like to be specific.