Darn there’s my theory up in smoke then 
I’ve lived in northern CA all my life, and have never heard the term. I’m also glad I’ve never heard the term. 
Kids in NH and ME have also been using it for years, and they too pronounce it vay-kay despite the spelling.
People here in the United States often talk about taking a “mental health day.”
I don’t believe this is the actual purpose of a personal (leave) day. At least, no one has told me so. I think it’s meant to cover unscheduled chores, like waiting in line all day at the department of motor vehicles after you’ve forgotten that your license plate is expiring or waiting for the plumber after your basement’s been flooded or having to take your kid home after school closes early for extreme weather.