Can practicing karate stances (horse stance, bow stance, etc) and katas aggravate herniated discs or are they safe to practice?

Depends: Certain positions particularly flexing and rotation are likely to irritate a pre existent dsc problem.

Shouldn't: If you do the horse stance and neutral bow stance properly you back is straight so no problems there. The rear bow and reverse bow stance may put a little pressure on your disc because your back is bent at an angle. It also depends on where your herniated disc is located. My comments are more for if you have a herniated disc in the lumbar region.