Learn French: Le vigile demande aux clients de patienter. - The security guard asks the clients to wait.

In French, the verb demander follows a specific pattern: demander à quelqu'un de faire quelque chose (to ask someone to do something). The preposition à combines with les to make aux clients, and de is strictly required before the infinitive verb patienter.

Why did they use patienter here rather than the standard verb attendre?

Patienter specifically means to wait patiently. It is the standard, polite verb used in customer service, announcements, and formal requests in French to sound more courteous, whereas attendre is just the neutral, factual verb for waiting.

Does vigile mean vigilante?

No, this is a common false friend. Un vigile is simply a private security guard or night watchman. If you wanted to talk about a vigilante taking the law into their own hands, you would use the word un justicier.

How could we apply the current lesson to say the guard asks them to wait, 'even if it means angering them'?

You can attach it directly to the main action: Le vigile demande aux clients de patienter, quitte à les fâcher. Since quitte à must be followed by an infinitive, we use the infinitive fâcher to express the negative consequence the guard is willing to risk to do his job.