불가하오니 is the same as 불가하니 or 불가하니까("because it is not possible/allowed") in meaning.
오 is a special suffix that makes certain verb endings sound more respectful, and thus suitable for when servants talk to their master for example, especially in the houses of high ranking officials in the old times.
In modern Korean, its usage is rare except in overly polite public notices as in your example.
Here are examples of the endings using 오 in the ancient times.
- 감사하옵니다 = 감사합니다.
- 바라옵니다 = 바랍니다.
- 그러하오면 = 그러면.
These are all but dead forms, so you shouldn't try to use it yourself.