How do I get a seat on the board of directors?

Do I just call or email the CEO and mention that I am a shareholder who wants to become a director?

You could certainly try!

But, no, your few shares themselves won't do it except be a nice story that the executive team and board might find amusing enough to entertain.

The CEO is typically appointed by the board, and may also be a board member, so you have to see what you can find out about every company. The board doesn't have to be multiple people, the single member on the board and the single member on the executive team can be the same person. For publicly traded companies this is uncommon, but the little known secret is that the market doesn't actually care. You might find yourself in a position where you get on a board. Being a board member for companies that need board members is actually a pretty lucrative trade. The level of independence and autonomy would be your niche, as there are "nominees" that dont really do anything but help a company gain clout and confidence, and there are not-quite-nominees. But there are also activist investors, who really do want control of the company.

The people that are typically on the board and elected to be there are people in that profession and with a lot of shares.

The point is: you get enough shares to DEMAND a seat on the board. Board members are representatives of shareholders and generally "a few shares of a public company" is nothing. Zero. No One cares about you. This is like asking McDonalds to change the menu because you would possibly buy one of the new burgers. No, not relevant.

5% of the shares is a blocking minority - then you can DEMAND.

Otherwise you need to convince larger shareholders that you are a good proxy for them. Have fun with that one. "I don't mind sitting in board meetings" is as entitled as it gets (you want a board seat, this is not "I do not mind", that is required) without a LOOOOT of credentials that make someone else say "hey, that guy could be a good board member instead of the other guy we use for years".