In order to activate a feature via the client side we simply have to know the GUID of the feature.

If this is a site scoped feature this is pretty straight forward.

The following can be run from a script editor webpart;

https://gist.github.com/ryanjonhealy/7c3a16008eec07ef70e2777847b0d528

We can then use the GUID found with the next function

https://gist.github.com/ryanjonhealy/c09a24f277c5f537b715edbb9539c749

This is especially useful when running with a function that manages site creation.