Web Design agencies aren't really the right target audience you need to talk to when it comes to websites and carbon emissions.
The emissions don't come from websites, they come from computer hardware, such as servers, switches, hubs, UPSs, cooling equipment etc etc.
Any change in the carbon footprint of websites can't really be driven by website design agencies, it would be driven by data centres. There are lots of UK-based data centres that promote themselves as sustainable.
Not quite true.
The host has a lot to do with it, but so does the design of the website.
It all comes down to the amount of data needing to be transmitted and the amount of processing required to deliver and consume the page/site.
Sites with large images, huge DOMs, redundant or inefficient code require more energy to do these things so have a bigger carbon impact.
Now you'll tell me that some hosts are running on green energy. True, but there are still caveats.
The host is one part of the equation, but so is the network and end user device.
Even still, the most hosts who say they run on green are addressing the 'process' (operational) emissions, but still don't talk about the 'embodied' emissions (which are typically 80-90% of the total footprint)
This website (websitecarbon.com) is a useful tool. If you happen to know of two different sites running on the same host, plug them both on there and you'll see they have different footprints (regardless of whether the host is on green networks or not)
To answer
@Paul Kelly ICHYB 's very valid comment ...
My enquiry is because I am trying to find design and development agencies that understand this and are trying to help their clients build more sustainable sites, with data that allows them to report their footprints (for regulation, certification, client or BCorp reasons).
Why? Well, I have a tool that can do this and I'm looking for people to work with.
Oh, and way at the top of the comments,
@antropy yes, you're right. My bad. In this instance it doesn't matter too much, probably what matters more is either a) how sustainable minded the owner of the website is (some business owners are trying to set good practice and leadership by minimising carbon footprints across all parts of their business) or b) whether the website owner is delivering products/services to clients in regulated sectors who have to report their 'Scope 3' emissions ... which include their suppliers' websites.
Hopefully that answers the questions above. (and prompts more??)