It does create an odd situation in the senate about voting him out.

Sounds like there’s a bunch of unresolved constitutional law questions there too. There’s apparently a literalist reading of the Constitution that the Vice President should preside over his own impeachment trial. Normally, the Vice President presides over the Senate. The Constitution explicitly says says that the Chief Justice rather than the Vice President presides if the President is being tried; but has no special exception in the text for if the Vice President himself is being impeached.

https://en.wikipedia.org/wiki/Vice_President_of_the_United_States

In their capacity as president of the Senate, the vice president may preside over most impeachment trials of federal officers, although the Constitution does not specifically require it. However, whenever the president of the United States is on trial, the Constitution requires that the chief justice of the United States must preside. This stipulation was designed to avoid the possible conflict of interest in having the vice president preside over the trial for the removal of the one official standing between them and the presidency.[49] In contrast, the Constitution is silent about which federal official would preside were the vice president on trial by the Senate.[13][50] No vice president has ever been impeached, thus leaving it unclear whether an impeached vice president could, as president of the Senate, preside at their own impeachment trial.

Then the process involves a vote in the Senate. On two occasions in the past, the Senate has in fact tied:

https://theconversation.com/the-senate-has-actually-tied-in-an-impeachment-trial-twice-130939

The Senate has actually tied in an impeachment trial – twice

The Vice President…holds a tie-breaking vote in the Senate.

https://www.legbranch.org/can-the-vice-president-vote-in-a-presidential-impeachment-trial/

In other words, the Constitution designates the Vice President the President of the Senate and gives the Vice President the power to cast tie-breaking votes. Yet the Constitution only designates the Chief Justice the presiding officer in presidential impeachment trials. The Constitution does not designate the Chief Justice the President of the Senate, nor does it give the Chief Justice the power to cast tie-breaking votes. This suggests that the Vice President retains the power to cast tie-breaking votes in presidential impeachment trials.

For bonus points, Vance is a lawyer, and I suppose could represent himself as well.

Anti-Vance Senator: “And, Mr. President, was that the point where the Vice President committed the treasonous crime requiring his impeachment and removal from office?”

Trump: “Yes, and…”

Vance (acting as Vance’s attorney): “Objection!”

Vance (acting as presiding officer over Vance’s trial): “Sustained. The court orders the statement to be stricken from the record and, further, observes that the witness is a whiny bitch.”

Vance (acting as holder of tie-breaking Senate vote): “Given that there seems to be a lack of evidence, I’m just going to come out right now and say that I don’t think that I can vote against the defendant.”

Vance (acting as defendant): “Woohoo! Score!”

EDIT: I don’t actually know whether the presiding officer when the Senate is acting as a court uses regular judicial procedure — could probably go dig up transcripts of past impeachments to find out — so the above phrasing may not be correct, but it’s still a pretty zany hypothetical.

EDIT2:

Trump: “This trial is a farce!”

Vance (acting as presiding officer over Vance’s trial): begins slamming Senate gavel “Order! Order! The witness will be silent!”

https://en.wikipedia.org/wiki/Symbols_of_the_United_States_Senate

The gavel deteriorated during the 1940s, and in 1952 silver pieces were attached to try to limit further damage. However, in 1954, Then-Vice President Richard Nixon pounded it during a heated debate over atomic energy, and it completely came apart. Officials wanted to recreate the gavel exactly, but not enough ivory was available commercially; Senate officials therefore contacted the government of India for help in sourcing the correct amount of ivory. On November 17, 1954, the Vice-President of India, Sarvepalli Radhakrishnan, presented the assembled Senate with a replacement gavel, which is still in use today.

Senate gavel comes apart again

Vance (acting as presiding officer over Vance’s trial): “Damn. Finding more elephants these days is going to be a pain.”

Trump: “I’ll say whatever I feel like!”

Vance (acting as presiding officer over Vance’s trial): “The sergeant at arms will remove the unruly witness!”

Jennifer Hemmingway: “Let’s go, Mr. President.”