Should answers be allowed to restrict which domains they can be run at? Should they be required to be runnable on all domains?
In my opinion, this constitutes outsourcing the real answer, which is a standard loophole. This is because it outsources the fetching of data from the website to the browser that ran the code.
I'd like to address the two "yes" arguments that were presented by Nathan Merril:
It is specified in the post
They aren't being used to store bytes (on code-golf or similar challenges where byte-count matters)
This doesn't make much sense to me. If a question explicitly allows something, it can be allowed, but this Meta post is tagged as policy, which indicates that we are looking from a general consensus in absence of any specification by the OP.
The definition of "storing bytes" is a little bit sketchy here. Even if the environment is not being used to store program code, it could still be storing data, and it is indirectly saving bytes by removing the need for a GET request or similar to fetch the data from the website. The code-golf standard loopholes are there to tell us what bytes are acceptable to save.
I'll finish with the ever-so-popular advice on PPCG: golf your code, not your IO format. Allowing code run at a specific domain feels too much like golfing the IO format to me.