I am looking for a way to create website (GTM) campaign tags for our email campaigns so that we have data in marketing cloud on how our customers are behaving on our website.
To have this data available in Marketing Cloud you need to have the "Google Analytics 360 Integration" purchased and set up in Marketing Cloud. In order to do so, get in touch with your account executive and read the following documentation: Google Analytics 360 Integration for Marketing Cloud
If you don't need the dashboards with Google Analytics Data to be available in Marketing Cloud, but just like to track everything coming from Marketing Cloud in your Google Analytics instance, you could request "additional email attributes" feature via a support case. This enables you to add your utm-parameters to emails in the email properties.
There is also a feature, called "Web Analytics Connector" that sets these parameters automatically in the background. This feature also needs to be activated by support: Web Analytics Connector