Criminal charges for what exactly? The SEC exists by authority of Congress, and their purview is defined by Congress, just like every other regulatory body.

The SEC even in overt insider trading cases doesn't pursue criminal charges -- they refer it to the Justice Department.

Congress currently is exempt from insider trading that comes about due to executing congressional duties. They would have to pass a law subjecting themselves to it.

> The SEC can still pursue criminal charges

The SEC can refer apparent criminal violations to the Department of Justice, which is the only federal agency which can pursue criminal charges.

The SEC is not a law enforcement agency. They can and do partner with the FBI but there's a separation of powers problem with that.