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The ball is hit to right center field, the runner misses first base while rounding it and goes on to be safe at second.
We tell the umpire about the missed base, and instruct our player to tag them on second, which is done while at second.
The umpire calls safe, then later said if we went to the mound and then threw to first, it would be out. I know that is one way, but can we appeal and tag the runner at second as well?
What the umpire is referring to is the appeal play, which is the one correct way to put out a runner who misses a bag. However, they may, or may not, be confused as to what actions constitute an appeal play.
From the MLB Rulebook (2021), 5.09(c):
Appeal Plays
Any runner shall be called out, on appeal, when:
With the ball in play, while advancing or returning to a base, he fails to touch each base in order before he, or a missed base, is tagged;
From this article, it shows two methods of appeal - live ball and dead ball. The rulebook does not specify when the appeal must happen; though dead ball is the more common method (and what is typically thought of as "appeal") for this kind of out, it is not the only one. (In fact, probably the most common appeal play is that where the player does not tag up after a fly ball - lining into a double play, for example - and that certainly happens when the ball is live in most instances.) Not all leagues support "dead ball" appeals; MLB does not, for example, but it is more common to be permitted at lower levels to avoid complications. I will assume only "live ball" appeals are permitted here, as otherwise simply informing the umpire of the desire to appeal should have sufficed.
What your question does not make clear is whether or not this occurred before or after time was called on the previous play. If it was a live ball situation and the play was still ongoing, then the player should simply have been ruled out so long as the umpire felt an appeal was being made (tagging a runner on a base should suffice, but the player could use the word "appeal" to make it more clear - but it should not be required). If it were after time was called, the ball needed to go back to the mound first, and then be thrown to the missed base properly (meaning, without a balk). The pitcher could also throw to a player who could then tag the runner who missed the base, if they are still on base; see for example this how-to. The rules do explicitly state "he, or a missed base, is tagged", and so tagging the player should be sufficient.
