There have never been [binding] vice presidential primaries in the United States, and even presidential primaries are a relatively recent development.

[EDIT: New Hampshire did hold a non-binding vice presidential primary from 1952-2008. If anyone is aware of other states with similar contests, I would be interested to hear about them. This contest was eliminated beginning in 2012.]

Until 1804, no one ran to be the Vice President of the United States. Rather, the Vice President was originally the person who received the second most votes in the presidential election. Unsurprisingly, this odd system soon resulted in the election of a President and Vice President of different political parties in the election of 1796 (President John Adams was a Federalist, while Vice President Thomas Jefferson was a Democratic Republican). This uncomfortable arrangement was changed by the 12th Amendment to the U.S. Constitution, which made the Vice Presidency a separately elected office. The election of 1804 was the first election to follow the newly revised system.

But there were still no primary elections. In the early years of the United States (before the 1830s), members of state political parties would caucus to select their presidential and vice presidential candidates. Beginning in the 1830s, parties began holding national conventions to select their candidates. State parties would choose delegates to send to the national party convention who would vote to nominate candidates for both President and Vice President on behalf of the state party. These candidates would then run in the general election, decided by the Electoral College.

National political parties didn’t begin to hold presidential primaries until the early 20th century, and even then many of the primaries were non-binding. Regardless of the outcome of the primaries, party delegates could choose whoever they wanted to be their presidential candidate. But there were still no primaries for Vice Presidential candidates; delegates at the party conventions continued to choose the candidate for Vice President (though the preference of the presidential candidate was often a consideration in this selection process).

It wasn’t until the 1970s that the national Democratic and Republican parties made presidential primaries binding, by which point it had become standard for presidential candidates to choose their own running mates. The system we have in the United States now - primary elections to choose a presidential candidate with that candidate choosing a vice presidential running mate - is only about 50 years old.

It is worth noting that the U.S. Constitution has nothing to say on the matter of how presidential and vice presidential candidates are chosen. The primary system as it exists now is entirely a creation of the major American political parties and follows rules created by those parties. As a result, parties often have slightly different rules for how they nominate a candidate, and the specific rules of a primary/caucus can vary from state to state based on the rules of the state party.