(…) and they were getting stronger with/for every day
What's the difference/the correct one to use?
With is correct here because it indicates the "getting stronger" and the "every day" are occurring simultaneously, or with each other. For is incorrect here because it would indicate that whoever/whatever is getting stronger is doing for the purpose of "every day," which is nonsensical.
However, the preposition could be dropped altogether with no loss of meaning:
The correct word here is with. I would not use every with it, however, but would phrase it like this:
If use of "every day" is important to you, you could phrase it this way, without any preposition: