Our company provides two weeks paid vacation a year. We have no sick time. Can a salaried person take additional days off and get paid? How frequently?

I believe that would depend on the specific laws in your state.

I've been a salaried employee for over 20 years and though the company doesn't actually list sick days, I've never been docked any of my salary when I had to take a day off.

That doesn't mean there are not laws that would dictate what is allowed.