How do I answer the question about the number of shares in an S corp at year end. We don't issue stock

If you never issue stock, then check your Articles of Incorporation. I think corporations have to issue stock (like LLCs issue "units") when created. It may not change and may not really matter, but the incorporators should own a share or a million shares or something.

I believe there are jurisdictions (states) that allow "non-stock corporations" BUT I also believe there are only a handful, if not just a very few.

Regardless, this type of corporation is NOT generally allowed to distribute income to its shareholders (owners) because they have none. As a result, the only form of income that can be received is reasonable compensation for services performed by, for example, a board member, or other employee of the corporation.

You really do need to consult with local tax and/or legal counsel.