On the Vita, PSN accounts are tied to the memory card as well as the system itself.
To swap the system you have to deactivate it first and reactivate it logged into the new PSN ID. It's best to just put the Vita through it's restoration process instead of doing it on Sony's website if you want to go back and forth between ID's because Sony only allows you to do it once every 6 months via the website.
You also have to format the memory card. Sorry, there's no getting around it. You can back up your save data on your PC using the content manager. When you start the Vita restoration it will ask you if you want to format the card. NOTE: You cannot transfer digital games to the new ID, they are permanently tied to the ID that purchased them. It should also be noted that DLC doesn't transfer between regions even though the games themselves are region free. If you have a physical copy of the game, you have to buy DLC from whatever region's PS Store is used by the region you're playing the game under. I have to buy Japanese DLC under my Japanese PSN account and can only use it with the game when logged into that ID.
Finally, set up your Vita again with the new ID. You have to watch that opening movie again and set the date/time.
If you plan on switching between regions often, it's best to just have separate memory cards for regions. I have one for the US and one for Japanese PSN accounts. Then all I need to do is restore/re-set-up the Vita when I want to switch and pop in the correct card.