How would you go about measuring the carbon footprint of a website?

not answering the question: in many situations it is likely that the direct carbon footprint of the website is completely insignificant compared to the indirect carbon footprint of the economic activity caused by the website.

e.g. does the website encourage people to buy more amazing kitchen gadgets, or fly more often to exotic yet authentic holiday destinations, or to meet sexy singles near you who are interested in starting families? that's where the real carbon footprint is going to be

Maybe you could get some ideas from https://www.websitecarbon.com/