From YPPedia
The Apply for Citizenship button appears in the Island tab in the Sunshine widget

Citizenship refers to the act of becoming a citizen of an island of a pirate's choice, given that this island is colonized.

To apply for citizenship, the pirate must be standing on open ground on the island, and have selected the island information tab on the sunshine widget. Then they must click on the Apply for Citizenship button. After confirmation, the pirate is immediately welcome to the island by the current Governor, added to the island's total population, and the island's dock area will be set as his/her new home.

The confirmation dialog.

See also