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We had a runner at third. The batter popped up into foul territory near the first base dugout. Both the catcher and pitcher went for it. The catcher made the catch, but there was nobody to cover home plate, so our runner ran from third and scored. What would this be considered?

This qualifies as a sac-fly per the following:

§10.08 (d) Score a sacrifice fly when, before two are out, the batter hits a ball in flight handled by an outfielder or an infielder running in the outfield in fair or foul territory that
(1) is caught, and a runner scores after the catch

team mom wrote:This qualifies as a sac-fly per the following:

§10.08 (d) Score a sacrifice fly when, before two are out, the batter hits a ball in flight handled by an outfielder or an infielder running in the outfield in fair or foul territory that
(1) is caught, and a runner scores after the catch

It looks like according to that rule, the ball has to be caught in the outfield. The ball was nowhere near the outfield. It didn't even make it half way down the first base line.

good point.

... that is always a challenge when responding to board requests and not seeing play live. .. sometimes we miss the obvious..

If it was clearly not an outfield fly (fair or foul) then no sac fly. .
( btw - i do not recall if/where MLB rules ever specify precisely when infield ends and outfield begins)

none the less if in scorer judgement not an outfield fly (fair or foul) , then you are correct not a Sac fly, just a pop out, (if dropped in fould it can be an error)

I would then score it a pop out on batter, and runner advance on the putout.

not a stolen base because the runner at third had the benefit of a putout being made on the batter.

no error because the failure to cover home plate by anyone on the defensive team is not a mental mistake that is counted as an official fielding error by the rulebook.