The dust-ups usually concern an employee requesting a vacation day that would result in the employer having to replace that employee and paying overtime to his/her replacement.
That would certainly "unduly interfere with the operation of the Authority."

here's the situation...an employee asked for a vacation day that would require our employer to pay overtime for someone to replace him. They are denying his vacation request BUT they will allow him to use a PTO day. Regardless of which type of leave they grant, it will necessitate paying someone overtime to cover his shift (he works solo on third shift that particular day). The reason that they cite for denying his vacation request is the "undue interference", claiming that paying another employee eight hours overtime is a 'hardship'.
I don't get it. What's the difference between one paid vacation day and one paid time off day?

The real reason that they are denying his request is because some of the PTO days can be taken with a one-hour notice (similar to sick days that we had prior to the aforementioned brilliant HR director). So they pull stunts like this to try to get employees to use up their PTO days before their vacation days so that employees can't save their PTO days until later, toward the end of the year.
I'm still not seeing it. You're still getting the day off. You just have to plan ahead a bit for vacation days later in the year. But you mean to tell me that come November you can't get one vacation day by asking for it the day before?

I'm trying to find a "legal" definition/interpretation of 'unduly interfere with the operation'
If it's not defined in your CBA I doubt that you'll find it in a statute or case law but other responders may have a different view.

I've done some research and the only thing that I have found that comes close is language in the ADA concerning workplace modifications.
That has nothing to do with this.

Is there something out there that I could possibly use?
Probably not. But you wrote that you are an officer in the union. Does your union have an attorney or access to an attorney? That would be the person to talk to.