Well, I see two possible ways for doing this.
First: Try to convince your client to launch a very simple powershell script for activating the solution throughout the mysite web application. You can give the script to their administrators for them to analyze and run if they're set on it.
Second: If you're REALLY set on not using powershell. Edit the profile page and add a script to it. This script will run each time and try to activate the feature in the user's stead. It's not really a very nice solution imo.
The REST URL you can use for activating the feature is this one:
Additional information can be found here: http://msdn.microsoft.com/en-us/library/office/jj245237(v=office.15).aspx