JEJEJEJE it is messy.
As many people know I think of vinira as a VERY slowburn and they don't date until years later, maybe even after the time skip.
But long story short: Emira confesses first because she's a flirt but also because Viney is probably the first real crush she ever had and she tries to do better.
She gets rejected because Viney's first impression of Emira is... bad. For her Emira is still this pretentious jerk girl who makes jokes and takes nothing seriously.
After that rejection Emira changes A LOT (at this point Amity is missing, and maybe she didn't change, she just started to grew older) so this new version of her makes Viney confused and of course, protective, and of course, other feelings start to bloom once she gets to know big sister Blight.
It is a big etcetera of things that happen between them in these months between labyrinth runners and Belos' defeat, they only get to work in their feelings after the final (I like to think they become very close working together at the healing track), but yes, that's basically how it goes.
A note to add: Viney also gets rejected but not because Emira doesn't like her jfjsjf, Emira is way too fragile and is afraid to hurt her because of that, also Edric was all wound up and she didn’t had mind for anything more than her brother 😭