To date, no sitting Vice President has ever been impeached, on account of the role having very little in the way of actual duties the VPOTUS could fail to do. The closest to this crisis occuring did occur during the Nixon Administration, when Vice President Spiro Agnew resigned (for matters unrelated to Watergate) leaving the position vacant as the likelyhood of Impeaching (and convicting) Nixon became a more political reality. The then Speaker of the House (a Democrat) realized he might not have the votes on articles of Impeachment to get charges to the senate with out Republican Support and that without the office of the Vice President filled, he was next in line of Nixon was convicted in the senate and that making such a move would make it look like a political coup (Nixon had won the previous election with 49 of the 51 states (D.C. gets votes in the Electoral College)). The Speaker discreetly pushed to get a replacement Vice President and told the White House he would use his party connections to ensure then Minority Sentate Leader Gerald Ford get confirmed by the Senate if Nixon Nominated him (Ford would become the only person to become U.S. President without having been elected to Vice President or President by the voters).
The entire period of time that the nation was lacking a Vice President during the Nixon Presidency lasted two weeks, so if in present political climate Pence was impeached and convicted before Trump, Trump could easily nominate Pence's sucessor, especially if said successor was backed by the Republican majority (incidentally, Pence was in the Senate prior to becoming VP, so it would be rather difficult to get his many close collegues to vote against him in sufficient number to convict. Although you wouldn't know it, most of the legislature, especially the Senate, act like the sheep dog and the coyote from that Looney Tunes cartoon. They are vicious enemies, but only while "on the clock" as it so war, and many long term Republicans and Democrats are good friends when not in the public eye. Lest the voter see them being friendly!).
Additiionally, per constitution, if someone is 3rd in line or below assumes the office of President, they are legally only "Acting President" and while they have full force of the office, politically the term notes that they do not have the political mandate of office and they may be greatly limited by the people. And if the office is assumed by a Legislative member, they must be willing to vacate their seat in Congress as part of the assumption of office, and will serve until the end of the term (right now, there's less than a year before the election, and 13 months before the next term begins... the Speaker of the House rarely will have a political future after leaving that office in this manner as typically to get to that position, one has to have been re-elected by their district over the course of several elections... which means they are typically in very safe districts and often on a far aligned spectrum of their party's ideologies. To my knowledge, I do not know of any one getting elected president after serving as Speaker (and the founders viewed the Speaker as the most important political role in the nation, though clearly that didn't happen.).
One interesting aside more: The hypothetical impeachment of the Vice President actually opens up a major constitutional crisis because while the Constitution does say the judge of the Senate trial must be the Chief Justice of the Supreme Court, it's rather mum on the judge for everyone else. This hasn't been a problem so far as most impeachments... but this is one of the few constitutional duties the Vice President does have (though for lesser Impeachments, it's normally the Senate Pro Tempore... who's always of the Vice President's Party and is really acting as President of the Senate when the Vice President can't actually sit in the chair himself (which is often). So the crisis is generally "Who is the Judge for the Veep's Impeachment Trial?" which has no legal answer as of yet because U.S. courts are constitutionally banned from hearing hypothetical cases. The running bust guess is that it would be the Chief Justice, like the President, but we don't know until it actually happens.