In NYT crosswords, “rambunctious NYT” usually points to a word like “wild,” “rowdy,” or “boisterous,” so the clue is simply asking for a synonym that fits the puzzle’s letter count and crossings.
The phrase call to whomever NYT crosswords typically refers to a generic shout you'd use to get someone's attention, such as "HEY YOU!" or "YO!" Therefore, the clue is referring to a casual phrase you'd yell rather than a literal phone call.