He joins Council members Clay Yarborough, Lori Boyer and Reggie Brown as those who entered the race for the leadership role.
Crescimbeni said Wednesday he "had every intent" on entering the race, but held off after his three colleagues announced last year that they would run.
He said he had not planned to announce in 2012 and waited to "see what would happen" in terms of which candidates had earned pledges of support among Council members.
Crescimbeni said he did not anticipate having one-on-one meetings with other Council members seeking support, as many do, and would let his record speak for his candidacy.
He served two Council terms representing District 1 from 1991-99 and served as vice president during the 1997-98 Council year.
He was elected to Council again in 2008 during a special election for the At-Large Group 2 seat and was re-elected in 2011.
In his letter, he said his more than 12 years of service has convinced him that "the Jacksonville City Council is capable of great success for taxpayers when we focus and work together."
Asked about the difference between his run for vice president now and the first time, he said "I have a lot more experience."
He said the past year has included numerous examples of his leadership style.
His letter references his chair position of the Council Finance Committee and efforts to save the Mayport ferry, oversee the Clerk of Courts office, restructure the Police and Fire Pension Fund board and save taxpayers $1 million through a waste-contract extension, which was approved Tuesday by Council.
Crescimbeni, a Democrat, said it is possible that partisan politics could happen in the Republican-dominated Council but "hope I would have the support of a lot of D's and R's." He said he had not seen party politics "creep" into Council elections and always had voted for who he thought was best.
Of the 19 Council members, 13 are Republicans and six are Democrats.
He said he had not provided a commitment to other candidates before his announcement.
According to minutes of several Council member meetings, Yarborough and Boyer have received support while Brown has indicated he has withdrawn from the race and would support Yarborough. Calls to Brown's office for confirmation were not returned by deadline.
According to the minutes, Council members Robin Lumb, Ray Holt, Stephen Joost, Don Redman, Doyle Carter and Brown have indicated they would support Yarborough. Warren Jones has indicated support for Boyer.
The Council vice president often, but not alwasys, is elected Council President the next year.
Council member Bill Gulliford is the current vice president.
Council elections for president and vice president are typically held in late May. The Council year begins July 1.
City Council member John Crescimbeni has entered the race to become the 2013-14 Council vice president, according to a letter he sent Wednesday evening to other Council members.