Either one could be correct, and there could be three different meanings.
Even though "bio and request" are nouns, they can be used as noun modifiers, or noun adjuncts (Wikipedia). This means they are acting as adjectives in this sentence.
If "bio and request" is a noun modifier phrase, then "the bio and request structures" refers to two structures. One of them is the bio structure and the other is the request structure.
"The bio and request structure" would refer to one structure, the bio and request structure.
Additionally, it's possible that "bio and request" is not a noun modifier phrase.
"The bio and request structure" could refer to two things, one is the bio and the other is the request structure.
Because it's grammatically ambiguous, we have to infer the meaning.
"He was eating the dog and cat food" is similar example. The likely meaning is that he was eating food for dogs and cats. However, it's also possible he was eating a dog, and food for cats.