There was never any standard practice not to renominate a VP to a second term. There were other issues at play. First, there initially was no distinction made between President and Vice-President on the Electoral College ballots, leading to awkward situations in the 1796 and 1800 elections that were consequently prevented thereafter with the ratification of the 12th Amendment in 1804.

Second, there was no process to fill a vacant Vice-Presidential seat until the passing of the 25th Amendment in 1967, so every time a President died and the VP was elevated to the Presidency, or the VP died or resigned, then the VP seat was left vacant until the next election.

Third, there happened to be a number of deaths of Presidents and Vice-Presidents that prevented the same ticket from running again four years later.

Fourth, there happened to be occasional political issues which prevented the same ticket from running again four years later.

One or more of these issues affected most elections before the passage of the 25th Amendment. When none of these issues were at play, the VP was re-nominated.

George Washington: John Adams served for all eight years as Washington's Vice-President.

John Adams and Thomas Jefferson: During the 1796 and 1800 presidential elections, the VP was simply the electoral college runner-up. This awkwardly made Adams' VP his 1796 political opponent, Thomas Jefferson. And then in 1800, Aaron Burr was Jefferson's "running mate" but because they tied in electoral votes, the election was thrown to the House. Instead of stepping aside, Burr challenged Jefferson for the Presidency, but Jefferson was ultimately appointed President. The 12th Amendment was then ratified to fix the issue, with a distinction now being made between POTUS and VP on the electoral college ballots. Due to the fallout from Aaron Burr not conceding the Presidency to Jefferson in 1800, and due to the fall-out from Burr's duel with Alexander Hamilton in the summer of 1804, Aaron Burr was not re-nominated by the Jeffersonians for re-election as VP under Jefferson.

James Madison: George Clinton had been VP during Thomas Jefferson's second term in office, so he was a holdover, but then he died in office. The office was vacant until Madison's second term, and then second VP Elbridge Gerry also died in office.

James Monroe: Daniel D. Tompkins served for all eight years as Monroe's Vice-President.

John Quincy Adams and Andrew Jackson: John C. Calhoun had political differences with Adams over the Tariff of 1828 so he decided to abandon Adams in Adams' re-election bid that fall. Calhoun ran for re-election as VP but threw his support behind Adams' rival, Andrew Jackson. Calhoun then was re-elected VP, but under Jackson, of the opposing party. John C. Calhoun subsequently had political differences with Jackson during the Nullification Crisis of 1832, and resigned, so he could be appointed to the Senate by South Carolina (the state causing the Nullification Crisis). Jackson then ran for re-election with a new VP candidate, Martin Van Buren.

Martin Van Buren: His running mate Richard Mentor Johnson was controversial because he was openly in a relationship with a slave (and would have married her but was prevented to by law) and fathered two children by her, and raised those children as his own, to be free upon adulthood. The South didn't like the implications, so he fell short of the required electoral votes to become VP when faithless electors from Virginia and other Southern states refused to vote for him, despite Van Buren handily winning enough electoral votes as POTUS to become President. Johnson then only got into office by a vote in the House via the 12th Amendment. Van Buren preferred Johnson to stay on for his re-election, but the Democrats didn't renominate him. Johnson still had enough supporters that they were able to prevent anybody else from getting enough votes to gain the VP nomination, either, and so Van Buren ran for re-election without a running mate. He lost.

John Tyler: Having succeeded to office, he did not have a VP during his first term, and wasn't renominated by the Whigs anyway, who had expelled him from the party.

James K. Polk: Ran for POTUS on a promise to serve only one term, and kept that promise.

Millard Fillmore: Having succeeded to office, he did not have a VP during his first term. He failed in his bid to be nominated for POTUS by his party anyway.

Franklin Pierce: Failed in his bid for re-nomination by the Democrats and they nominated new POTUS and VP candidates.

James Buchanan: Probably would have suffered the same fate as Pierce, so he did not run for re-election. Once again, the Democrats ran new POTUS and VP candidates.

Abraham Lincoln: Out of concern he would lose re-election, the Republicans nominated a Democrat from the slave state of Tennessee under the banner of the National Union Party.

Andrew Johnson: Having succeeded to office, he did not have a VP during his first term. He failed in his bid for "re"-nomination by the Democrats in 1868.

Ulysses S. Grant: His first VP, Schuyler Colfax, announced halfway through Grant's term that he would not run again in 1872, hedging his bets that Grant would retire at the end of his term and/or be unpopular enough that Colfax could get the nomination in 1872. It didn't work. Grant was re-nominated, won with a new running mate, and Colfax was left out of Grant's second administration.

Rutherford B. Hayes: After the controversy of the 1876 election, Hayes pledged not to run for POTUS again and kept that promise.

Chester A. Arthur: Having succeeded to office, he did not have a VP during his first term, and he declined to seek "re"-election.

Grover Cleveland: His first VP, Thomas Hendricks, died in office. He lost his re-election, but then won his second term four years later with a new running mate.

William McKinley: His first VP, Garret Hobart, died in office. He won his re-election with a new running mate, Teddy Roosevelt.

Teddy Roosevelt: Having succeeded to office, he did not have a VP during his first term, so ran with his first running mate during his "re"-election.

William Howard Taft: His VP, James S. Sherman, died in office, so he ran for re-election with a new running mate.

Woodrow Wilson: Thomas Marshall served for all eight years as Wilson's Vice-President.

Calvin Coolidge: Having succeeded to office, he did not have a VP during his first term, so ran with his first running mate during his "re"-election.

Herbert Hoover: Charles Curtis was his VP, and he ran again with Curtis in 1932, but lost.

Franklin Roosevelt: John Nance Garner served as FDR's VP for his first eight years in office. In 1940, Garner announced his candidacy before FDR did, but then FDR announced he was running for a third term. Garner didn't drop out, and challenged FDR for the nomination, but FDR was handily re-nominated. He chose Henry Wallace as his new running mate, which was controversial with the conservative wing of the party, and only won 59% of the delegates at the convention (where FDR won 86% of the delegates for POTUS, compared to Garner's 5%).

For his fourth term, FDR stuck by Wallace, and Wallace won the most votes on the first ballot of the convention, but not enough to get the VP nomination. Party insiders had become aware of FDR's failing health and conservatives in the party were anxious of Wallace succeeding to office. Truman was advanced at the convention as an alternative, and won the VP nomination on the third ballot, defeating Wallace. FDR won the election, and Truman became VP.

Harry Truman: Having succeeded to office, he did not have a VP during his first term, so ran with his first running mate during his "re"-election.

Dwight Eisenhower: Richard Nixon served for all eight years as Eisenhower's Vice-President.

Lyndon Johnson: Having succeeded to office, he did not have a VP during his first term, so ran with his first running mate during his "re"-election.

In 1967, the 25th Amendment to the Constitution was ratified, which, for the first time, made it explicit that a VP who succeeded to the Presidency was the actual president and not just an "acting" President. It also, for the first time, outlined the process by which the VP vacancy could be filled, when a VP succeeds to the Presidency, or when a VP dies or otherwise vacates the office.

Richard Nixon: Spiro Agnew was Nixon's running mate in both of Nixon's successful Presidential campaigns. Agnew resigned during Nixon's second term when Agnew was being investigated for criminal corruption, and Gerald Ford became the new VP under the 25th Amendment process.

Every re-election campaign since has seen the POTUS run with the same VP they did during their first term, both in successful and unsuccessful re-election bids.

So it wasn't a matter of it being "standard practice" that Presidents sought out a new running mate. When the POTUS was successful in getting re-nominated, and when there was an actual VP occupying the office of VP, then the VP was re-nominated, too, except on the occasion where there were political issues at play, as happened under the re-election campaigns of Adams, Jefferson, Quincy Adams, Jackson, Van Buren, Lincoln, and Grant, and the third and fourth campaigns of FDR. Adams and Jefferson suffered from the initial problems of the Electoral College. Quincy Adams, Jackson, Grant, and FDR (during his third term) suffered from a disloyal VP. Van Buren, Lincoln, and FDR (during his fourth term) suffered from issues surrounding party politics that prevented their VP from being renominated, even though they would have preferred the VP to stay on.