You cannot use the GA API without a form of authentication, therefore connecting GA directly to your website without client login isn't possible.
As a workaround, you can use a middleware solution to pull data from the GA API, and then pull data from that midddleware to your website. Here is an example:
GA API ---> Google Sheets ---> Website
- Use the GA add-on to pull your data to a Google Sheet
- Import that data into another Google Sheet with IMPORTRANGE
- Publish the Google Sheet from 2 so it's publicly readable via API (for instance as CSV). You will be provided with a URL that looks like
https://docs.google.com/spreadsheets/d/e/{some-uid}/pub?gid=0&single=true&output=csv
- Fetch data from that URL from the website (eg with jQuery)
Note: you probably don't have to do step 2 if you're careful, and can publish as CSV your sheet from steo 1 directly: since it's a CSV export, people cannot get to the original spreadsheet so no security risk. But just to be on the safe side (eg you could publish the whole spreadsheet by mistake), I would do step 2.
As for which middleware you want to use, there's probably an infinite number of options :)