Is it true you can get in trouble for looking for a job while you're on a leave of absence or vacation? I can't remember where I learned of that but I think it was said that if you go away "to school" and they find out you were job hunting, you can face your demise
How would they know? What you do outside of work is YOUR business.
Some companies do this. For instance a tech company. If the techie applies to the competition and then they call the current employer for a reference, the current employer most likely will let the employee go. Or, I have read about people calling in sick but later seen on TV at a major sports event. Fired again. Or, on paid medical leave but Facebook photos reveal a surfing vacation in the Bahamas.
For a menial job like Kroger, I doubt they will fire the employee if they find out. A good store manager will let employees explore other opportunities.