Vice President Aaron Burr was indicted in his last year as Vice President in both New Jersey and New York. The crime was killing Alexander Hamilton in an (illegal) duel. His solution? Simply don't go to those states.
Eventually the charges were dropped on technical reasons. But it was nevertheless clear that no one considered it a particular problem for a sitting Vice President to be indicted in multiple states at a time, or even just one.
This bit of history is even mentioned in the revised DOJ opinion referred to in Putvi's answer. Ultimately that opinion decides that the comparison between a Vice President and President is apples-to-oranges nonsense—their relative levels of power, importance, and constitutional stature are radically different—, and no substantive inferences could be made from a VP's indictability to the President's.
Of course, this DOJ opinion is just the opinion of DOJ lawyers. It is not a court opinion, nor otherwise binding. It does strike me as well-written and logically strong, but it is wholly untested in a court of law (SCOTUS or otherwise). It is simply their advice on what the best course of action the DOJ can take is, given their particular function and powers within the US government and overall constitution. The main conclusion being that those functions and powers are inadequate for the DOJ to decide if it is constitutionally sound to indict a sitting President (that's a job for the judiciary), and so simply advises that they avoid that powder keg and never do so. The opinions actually do specifically hold that it might be constitutional in the right situations; they simply conclude they cannot and should not be the ones deciding if a particular case qualifies or not.
For a more definitive answer, you may have to wait for various court cases against President Trump and his administration to make their way through the system. In several of them the President's/White House's lawyers have argued for absolute immunity on all criminal matters, state or federal. The judges have, to date, seemed disinclined to agree; many seem straight-up shocked that this is being seriously proposed to them. But those cases are still in progress, so we can at best make random guesses at what the actual opinions will be, and it is even harder to know what will happen on the almost-certain appeals up to the Supreme Court.