This could occur, but it is highly unlikely. In the event that Pence is co-implicated and could actually be impeached, he will likely resign ahead of Trump. This will allow Trump to nominate a new VP that will require approval of both the house and senate, but it is unlikely that the Senate will vote against Trump anyways.
Fortunately, the 25th Amendment allows for the replacement of a VP even mid-office. And confirmation hearings would likely mire Congress for months if not all the way through 2020 - at which point Trump could just pick a new VP running mate.
Then, if Trump is impeached or resigns they are covered. If this were to occur it’s basically Agnew -> Ford -> Nixon -> Ford kind of situation.
That said, we’ve never had a successful presidential impeachment that required a a POTUS to leave office.
I also don’t believe it is possible to bring simultaneous impeachment proceedings against a President and Vice President. Even if the are “co-conspirators”, because of the nature of the crimes that allow impeachment proceedings to be brought.