Write up for losing a CAC? Supervisors discretion?

Can supervisors write people based on no written policy?

I'm dealing with a co-worker losing their CAC, essentially part of a new union OCONUS with no CBA as of yet.

Based on the previous post it originally started with the supervisor saying he must use annual leave to replace the CAC and not get admin leave. He eventually reversed this decision, but held his mindset - saying the employee caused the issue and it's no different than him being late and he would have to use annual leave.

But; he made a point to say he could write him up for 'negligence.' Are such things based on the manager's discretion? Never seen a CBA address such issues and there is no HR policy stating losing a CAC will end up in documentation.