Sir Vince, 74, returned to the House of Commons as MP for Twickenham in this month’s General Election, having been one of the highest-profile casualties of the party’s collapse in support in 2015.
Former health minister Norman Lamb has also indicated he is considering a bid for the leadership, but hotly tipped East Dunbartonshire MP Jo Swinson, 37, has ruled herself out, saying she will fight for the deputy leadership instead.
Declaring his candidacy on the Lib Dem Voice website, Sir Vince said he is ready to “work with like-minded people in other parties” to secure a second referendum on any Brexit deal, with the option to stay in the EU if the agreement on offer is not good enough.
He described Brexit as an “iceberg” about to hit the UK economy and said the party should “warn of the dangers ahead and the need for a new course”.
Despite the disappointment of Liberal Democrats securing only 7.4% of the vote and 12 seats in this month’s election, Sir Vince insisted that “the political winds are moving in our favour”.
He said: “There are big opportunities ahead. The Conservatives are in disarray and in retreat. The Labour Party outperformed expectations but complacently believes that ‘one more heave’ will see it into office.
“But an economic policy based on offering lots of free things lacks economic credibility and will be found out. Investing in infrastructure, rather than borrowing for everyday running costs is credible. There is a big space in British politics which I am determined that we should occupy.”
Sir Vince previously served as acting leader following the resignation of Sir Menzies Campbell in 2007, but declined to stand for the top job at that point, saying that an older candidate would not be electable because of “irrational prejudice about age”.
If elected, he would be the party’s oldest ever leader and the oldest leader of a major party since Sir Winston Churchill, who was 81 when he stepped down as Conservative leader.
Former energy secretary Sir Ed Davey has also said he is giving “serious thought” to standing in the leadership election.
Mr Farron shocked many Lib Dems by announcing days after the election that he will stand down when Parliament breaks for the summer next week.
A committed Christian who was repeatedly questioned during the election campaign over whether he saw gay sex as a sin, Mr Farron said his religion had made him a “subject of suspicion”.
Contenders to replace him must submit nominations from at least 10% of the parliamentary party - two MPs, not including themselves - as well as at least 200 members drawn from 20 or more local parties.
After a contest stretching over the summer, the new leader will be elected by party members in a postal ballot conducted by a single transferable vote in time for the start of the Lib Dem annual conference in Bournemouth on September 16.
During the summer, the new Lib Dem deputy leader - due to be elected later this month - will lead the party politically while president Lady Brinton will oversee internal party affairs.