Nope, each branch can appoint an independent counsel.
The independent counsel law was thrown out by SCOTUS after Billy Jeff's impeachment. They decided it was a violation of the separation of powers - only the executive branch has the ability to prosecute anyone. The only powers that Congress has are the power of the purse and impeachment - for anything else, such as lawbreaking by the executive branch, they must rely on the executive branch to prosecute itself...