I would agree with edralph regarding the ownership of the shares prior to exercise. Often, the company issues new shares when the employee exercises the option. The company just needs to make sure it has the authority to issue such additional shares (usually determined by looking at the Articles of Incorporation).
LLC's can have option plans as well, though they are usually called unit options instead of stock options. A unit is basically the same thing as a share of stock (talk to your lawyer to make sure your LLC is structured in this manner, it can vary). Unit options work very similarly to stock options, and are taxed in generally the same manner.
This is clearly oversimplified, you should discuss your particular situation with an experienced attorney so that they can help you craft a plan that fits your current and future needs.
Attorney, BrownWinick Law Firm
Until stock options vest, there is no need for real stock to even exist. So there is nothing to own. Immediately before stock options vest the company can either buy shares on the open market (if it is a public company) or it can 'create' stock after it has been authorised to do so by shareholders.
Typically an LLC doesn't issue stock (maybe even cannot issue stock) as the ownership mechanism is different from a corporation. So I believe stock options are not compatible with the ownership structure of an LLC.