The short answer is that the Vice-President does not get to cast any vote in case the Senate has to elect the Vice-President. My full answer is below.
In order to confirm my answer, I wrote an email to the United States National Archives and Records, which maintains the website The US Electoral College, including the page Presidential Election Laws, and received this response (emphasis mine; words in brackets original to the email):
A Vice-President voting to break a tie in a legislative vote under Article I is still voting as Vice-President, not as Senator.
The 12th amendment says that a "majority of the whole number [of Senators] shall be necessary [to elect the Vice-President]."
The 12th amendment affects Article II, not Article I, and only specifies a vote by Senator.
If you have more specific questions about Senate procedure, you'll need to contact the Office of the Secretary of the Senate.
Legal Affairs and Policy Division
Office of the Federal Register
National Archives and Records Administration
My question is that if the senate vote for VP and ties again, can current VP vote to break the tie?
No. The Vice President does not get to cast any vote, including a tie-breaking vote, in the Senate's election for that same office of vice president!
Read the XIIth amendment:
The person having the greatest number of votes as Vice-President, shall be the Vice-President, if such number be a majority of the whole number of Electors appointed, and if no person have a majority, then from the two highest numbers on the list, the Senate shall choose the Vice-President; a quorum for the purpose shall consist of two-thirds of the whole number of Senators, and a majority of the whole number shall be necessary to a choice.
Note well: a majority of the whole number [of Senators] shall be necessary to a choice.
It is the Senate of the new Congress that would elect the vice-president. Although the Vice-President is acting as the president of the Senate, the Constitution does not give him/her the right to cast a tie-breaking vote on this occasion.
Notice again the wording of the ThoughtCo quote:
If the Senate has failed to break a 50-50 tie for vice president, the Presidential Succession Act of 1947 specifies that the Speaker of the House will serve as acting president until tie votes in both the House and Senate have been broken.
Notice "50-50 tie." This clearly refers to the full Senate voting and tying 50-50. There is no mention of any tie-breaking vote by the VP because he or she does not cast a vote in this situation.
And that, fortunately, makes sense, because if the incumbent VP was one of the two VP candidates who tied in the electoral college vote, then it should not devolve upon him/her to cast a vote for himself/herself in order to break the tie-vote of the Senate.
(Last, the fictionalized TV show VEEP is wrong on several points regarding the whole electoral tie thing: NB that if the House doesn't successfully decide upon a president on the first vote, the House keeps on voting until it does get a majority vote. In 1800 it took 36 ballots before the House voted for Thomas Jefferson; of course that was pre-XIIth amendment, but the XIIth didn't change anything in that regard-its change was that electors no longer cast two votes for president but rather they now cast one for president and one for vice-president.)