I'm I'm not sure if I'm correct, but if I had to guess what Aya meant by the Santa comment, I would think she meant something like "Did I get here before the party ended?". You usually open the presents last and Santa is the one delivering them. And you can see that when Aya arrived her scarf and hair was messy, which I assume is because she ran there as fast as she could so she wouldn't miss the chance to see Mitsuki.