Learn Turkish: O zımba çalışmıyor, lütfen yeni bir tane al. - That stapler isn't working, please buy a new one.

In English, we use "one" to avoid repeating a noun, like saying "a new one". In Turkish, for physical, countable objects, we use tane (which literally means "item" or "piece") in the exact same way. Yeni bir tane acts as a pronoun phrase so we don't have to repeat the word zımba.

Does çalışmak apply to machines the same way "working" does in English?

Yes. Even though çalışmak is usually learned first as "to work a job" or "to study", it is also the standard verb for machines, electronics, and mechanical tools functioning properly. When a device is broken, you simply say it is çalışmıyor (not working).

Why use al instead of satın al for "buy"?

Satın almak is the specific verb for "to purchase." However, in casual, everyday speech, Turkish speakers overwhelmingly use almak (to take / to get) when the context of buying is obvious, very similar to how English speakers say "please get a new one."

Could I say şu zımba instead of o zımba here?

You could, but it changes the physical context. O zımba implies the stapler is either out of reach for both of you or was just mentioned. If you use şu zımba, you are actively directing the listener's eyes to a specific stapler they might not be looking at yet, or pointing out a stapler that is sitting closer to them than to you.