The DoJ and FBI are both part of the Executive Branch of the government.
Each branch of government, by default, assumes that the Constitution should be interpreted in its favor and will make all arguments to that effect and act in that understanding, right up to the point where the question actually makes its way to the Supreme Court and the justices decide otherwise.
The Executive branch has, long-since, made it public policy that the President is immune from prosecution while serving in office. While I don’t believe that this is written anywhere in the Constitution, I think it comes from a general understanding that a) the President can’t do his job if he’s constantly in court, and if people could simply take the President to court for everything he did that they didn’t like, then there wouldn’t be a day that a new case wasn’t filed, since his actions affects the whole country, and b) if the President can be convicted, then that’s giving the Judicial branch ultimate power over the President, which isn’t terribly kosher.
But so, Mueller is working as a subordinate in the Executive branch and so, regardless of whether this policy position would actually prove out if put to the test in the Supreme Court, he is still bound to uphold it.
That said, he can a) issue a sealed indictment that will trigger as soon as Trump leaves office, b) note him as an unindicted co-conspirator as part of issuing an indictment against someone else (e.g., Don Jr.), c) ask the Attorney General (in this case, Rosenstein) for an exemption from the current policy, on the basis of extreme circumstances, and/or d) issue a report to Rosenstein on “impeachable offenses”, which Rosenstein would likely share with Congress.
However, it’s incorrect to say that Mueller’s mandate is to investigate Trump. Regardless that the media and Trump have treated it that way, it’s not correct. As one person has noted, both Mueller and Comey were principally investigating Russian acts to influence the American election, and that investigation is what Trump is accused of acting to obstruct (which seems pretty curious if you think about it).
However however, that’s also not what Mueller is investigating. Here is the specific mandate that Rosenstein issued:
There is a further, secret, mandate that was issued along with this that further identified (we believe) four different individuals associated with the campaign, and specific criminal acts that they may have committed. On this document, everything other than Paul Manafort’s section (and possibly not all of that) has been redacted, so we don’t know who the other three individuals are. It would seem likely that Trump would not be one of them, given the aforementioned policy, but we do have some reason to believe that Mueller is investigating Trump’s obstruction of justice issue.
But, it should be noted, the only reason that we believe that Mueller is investigating Obstruction of Justice is because Trump’s lawyers leaked a document that was purported to contain the questions that Mueller had for Trump. It is conceivable (though, hopefully not likely) that this was wholly manufactured by Trump’s legal team in order to make it appear that Mueller wanted to meet with the President to ask about Obstruction of Justice, when in fact he might be asking Trump to tell him about Don Jr’s discussions related to the Russian lawyer, and whether Rick Gates had anything to do with Mariia Butina and the NRA. The fact that these latter questions were missing from the leaked list of questions make it likely that the document was at least edited down if not wholly fabricated.
Overall, there’s no short and easy explanation about Mueller’s mandate nor answer about what he is working on. But from an official standpoint, we can say that he was assigned to investigate specific individuals related to the campaign (likely including Trump’s children), but we do not know if that includes Trump. But he can investigate any person or crime that follows from those he encounters while pursuing his mandate, so that could stretch to include the President - except that Mueller will have to give Trump immunity by default until 2021.
And, it should be noted, it is quite interesting that Mueller has been indicting Russians as part of an investigation into the Trump campaign members. They should, by virtue of his mandate, not fall under his directive. That he ended up indicting them would imply that somehow the individuals in the campaign lead directly to those Russians.