“Has a sitting Secretary of State ever immediately become their party’s next presidential nominee, bypassing the sitting vice president in the very next election?”

Then yes, but mostly in the 1800s.

The clearest example is probably:

James Madison — Secretary of State under Thomas Jefferson, then became the Democratic-Republican nominee in 1808 even though Jefferson’s VP, George Clinton, was still vice president.

Another:

James Monroe — Secretary of State under Madison, then became the nominee in 1816 over VP Daniel D. Tompkins.

And:

John Quincy Adams — Secretary of State under Monroe, then won the presidency in 1824 instead of Monroe’s VP, Tompkins.

In that era, Secretary of State was widely considered the “heir apparent” office. The vice presidency was weak and often politically unimportant.

In the modern era, this basically has not happened when a sitting VP actively wanted the nomination. The closest modern analogue is:

Hillary Clinton becoming the Democratic nominee in 2016 while sitting VP Joe Biden chose not to run.

So:

Early republic: yes, repeatedly.

Modern party-primary era: essentially no successful example against an actively running vice president

(http://www.autoadmit.com/thread.php?thread_id=5864291&forum_id=2id.#49869688)

did tompkins ever try to get his party's nomination

No — Daniel D. Tompkins never seriously sought the presidency or mounted a real campaign for his party’s nomination.

By the time of the 1816 election:

Tompkins was serving as VP under James Madison.

James Monroe, the Secretary of State, was already viewed as Madison’s natural successor.

The Democratic-Republican congressional caucus overwhelmingly backed Monroe.

Tompkins actually ended up becoming Monroe’s running mate instead, serving as vice president from 1817–1825.

A few reasons he wasn’t a major contender:

He had financial and health problems.

He lacked Monroe’s national stature and diplomatic résumé.

The Secretary of State position was still seen as the premier launching pad to the presidency.

So this wasn’t really a case of a Secretary of State defeating an ambitious sitting vice president in a contested succession battle. Monroe was simply the established heir apparent.

(http://www.autoadmit.com/thread.php?thread_id=5864291&forum_id=2id.#49869693)

Specifically:

a sitting vice president actively seeking the nomination,

losing to a sitting Secretary of State,

in the modern primary era (roughly post-1968),

has not happened.

ABSOLUTELY DEVASTATING for "JD" Vance

(http://www.autoadmit.com/thread.php?thread_id=5864291&forum_id=2id.#49869700)