Is it possible?

To switch an existing character on a team for another character and keep your spot on the team without saves? xD I plan on having Rue adopted, but I really want to keep my spot in Ilex -and a better character to put in there lsdkfjasf-

Just askin', I saw someone do it once, but I dunno if there's a certain process to go through

Yes, that's permissible.

I've done it. You just ask to switch a character you have active with a character pending approval in the Profile Computer. Once the switch is approved the character can get approved and you are in, then your character being switched out goes inactive.