Is it possible for me to move in two animals on the same day? Say I go to someone's town and convince Tia to move in, am I able to go to another town on the same day and invite another villager to move in, or to the campsite for that matter?
Only one villager can move in per day, regardless of where they come from. The last villager to be invited will be the one that shows up and overwrites the others.
I think FireNinja is right about that letter only being sent when there isn't enough space in your town for the villager to put down a plot. But yeah, as has been mentioned, the second villager you asked to move in will overwrite the first. So you'll need to invite them on separate in-game days.