"They are not swimming because of the cold water."
Translation:Ze zwemmen niet vanwege het koude water.
"Doordat" is a subordinating conjunction that connects a subordinate clause to the sentence. "Het koude water" is not a clause. A clause is like a sentence. That is, it has a subject and a verb at least. What follows "doordat" has to be a sentence.
I think you could say (grammatically, at least), "Ze zwemmen niet doordat het water koud is." Here, "doordat" is expressing a relationship between "They don't swim" and "The water is cold." But I think "omdat" would work better than "doordat" in that sentence. "Doordat" expresses more of a direct causal relationship between the clause it introduces and the main sentence, I think. But that's a lengthier discussion, perhaps for another day.
In any case, here, since the lesson is about prepositions and you're given only "the cold water" and not a clause (like "the water is cold"), the correct answer will be a Dutch preposition, "vanwege," and not a Dutch subordinating conjunction (like doordat or omdat).