I understand that the Justice Department can’t charge a sitting president with a crime, but what could they actually do if a special counsel like Mueller found that it seemed very likely the president committed a crime? Would they send the evidence to Congress, or would it just sit around in a cabinet forever?
If the crimes were serious enough, they could defer prosecutorial actions (defer all aspects of prosecution, including an actual official criminal investigation and charges, if any were warranted) until the President was out of office, which would set up a scenario of the President trying to pardon himself, or possibly resigning in a Nixonian deal where the resignation is paired with the subsequent office-holder issuing a pardon, which might set up yet more investigations.
From what Mueller has said, and contrary to what Barr has offered, the entire structure of the Mueller report was a road map for Congress to use if they wanted to push forward with impeachment proceedings.
In that regard, what the Justice Department could do is offer the full, unredacted (or as much as can be not redacted) report to Congress, and offer to make members available for any formal impeachment proceedings to provide testimony.
Robert Mueller finally made a statement about his report today, 5/29/19. The statement was very short (~10 mins). And it was so universally covered, that I won't try to link to it.
According to the statement made by Robert Mueller today, special counsel cannot indict a sitting President nor can it produce a sealed indictment (an indictment secretly held until a future date with the purpose of waiting out a temporary immunity while not running up against a statute of limitations).
Robert Mueller stated that this DOJ guidance is due to the fact that indicting a sitting President would be unconstitutional. Because this question has never been taken up by SCOTUS, this is currently the most authoritative answer on the subject.
Referring it to Congress is the only course of action available to a special counsel.
A point which maybe worth noting is that the special counsel Mueller's office did not make a Congressional referral. While Ken Starr's report was referred to Congress. Obviously people will argue that this is due to them having different offices and responsibilities (Starr was an Independent Counsel while Mueller was a Special Counsel). But Mueller did have the option of making a referral.
A number of people have suggested that the language which Mueller used suggested that he was making a referral. But that is simply not the case.
He made a number of prosecution referrals during the course of the investigation. So making referrals outside of his office was within the scope of his powers.
He could have simply made the referral if he thought it was appropriate. He did not.