The Department of Justice, under the federal statute. Under the common law rule, it would be the person who would be entitled to the office if the writ were granted (which would be Joe Biden).
So unless a sitting administration asks for it for the new administration before they take office, it will never happen. Both of the ones with the right to request are the ones who would not do it (unless Biden gets the desire to make his boss mad).
It seems to me the Framers created the 18th century version of an MS Windows "security hole". I don't claim to know about Obama's BC, birth location, or eligibility. However, if he secretly isn't truly eligible, he pulled off the biggest con job in history, and he owns the keys preventing discovery to prove the case for the people.