Well... For us the issue was this:
We had a chat-function on our page. It seems that the js on the chat-function was overriding the “add to cart”-button.
The “add to cart”-button was functioning on our front page. Then the chat was set in motion, and that seems to have deactivated the js on the “add to cart”-button further “down” on our page.
The solution for us was to remove the chat-function from the store…
Ideally, I would like to have the chat there as well, but now we have the chat on our main page and a functioning web store as well.
So I guess the quick answer is to check if there are different plugins or modules that do not work well together…