Is it obligatory to verify the legitimacy of a shop's activity—whether its owner is a disbeliever or not—before designing a paid sign for it, or providing application systems with information, given that the legitimacy of its activity is unknown?

The designer is not obligated to scrutinize the activity of the store owner (Ahmed), and whether he will sell something forbidden or permissible, as long as the designer is ignorant of his condition, even if the owner is a disbeliever. This is because the subject of the contract is a permissible service. The default ruling is the permissibility of dealing with disbelievers in buying and selling, provided that the transaction involves something permissible. This is evidenced by the Messenger of Allah (peace be upon him) buying food from a Jew.