"make a request" sounds totally normal in that sentence. They are talking about a formal request for someone to do something in a professional work environment. In casual conversations, however, it would be more common for people to talk about "asking" someone to do something. A mom wouldn't say to her child, "Oh, you want a ride to your friend's house? Go make the request to your father." ha ha. In such casual conversations, it will definitely more natural to say, "Go ask your father." :-)

Doesn’t it sound odd at all?

That sounds right to me! I might remove the "and" -- “The attorney general needs to make the request to the president, not to the vice president”