Do I have to issue shares for my S Corp when I am the sole owner?

So long as you are the sole owner you are the sole shareholder.  It's best to issue the shares to evidence your ownership but if you don't, you are still the sole shareholder.

Sorry to resurrect this old post, but a quick follow-up:  as the sole shareholder of an S Corp, if the S Corp did not issue stock, how could the shareholder calculate their S Corp basis, as required on line 28 of Schedule E (1040)?

Just issue yourself 100 shares for income tax purposes ... you can always "print" them out later if needed.