WAGO 750-841 - How can I upload temperature data to a website?

Welcome,

the question will probably seem trivial, but I am wondering how to upload the data from my controller to a website.
I would like the temperature measurement, or rather the results of that measurement, to be available to me at any time from anywhere I have internet access.

Well, that's different.
I access the internet via my brother-in-law's wifi network. The router is located in the house next door.
Is there a chance to connect his router to mine wirelessly without losing the internet? Then I would connect my WAGO to my router and I could continue to combine.

I think you have described vaguely what you already have.
If I understand correctly your brother-in-law has internet access + a router with wifi.
You have internet from him because you are in wifi range.
When you write "my router" do you mean the router you are yet to purchase?
If so then you need a router with bridge capability.
The downside is that you will be on the same subnet as your brother-in-law, which means he will have easier access to your networked devices.

If you're really close (90mb max between your devices), I'd recommend pulling a cable from him to you and plugging it into the wan port on your router. The connection between devices will be fast and stable, and you will be on another network protected by your router's firewall.
Robert.