Is it permissible for an employee to use the institution's internet for personal purposes in exchange for a fee, knowing that this use may include browsing social media websites?

There is no harm in subscribing to and using the internet at work if it is authorized by the person empowered to grant permission. What is mentioned is closer to a waswasah (a whisper from Satan), so one should disregard it and benefit from the internet in what is permissible, whether it is specific to work or not.