Learn Spanish (Latin America): El cliente necesita su recibo.

You can simply change the article (the "the" word) to match the person: el cliente for a man and la cliente for a woman. However, in many parts of Latin America, it is also very common to say la clienta for a female customer. Both are correct!

How do we know su means "his" here instead of "her" or "your"?

By itself, su can mean his, her, your (formal), or their. We translated it as "his" here assuming the male customer (el cliente) is getting his own receipt. If we were talking about a female customer (la cliente), su recibo would naturally translate to "her receipt".

If the customer just needed "some" receipt, would I use alguno or algún?

You would use algún. Because recibo is a masculine, singular noun, the word alguno drops its final -o and gains an accent mark when placed directly in front of it. You would say: El cliente necesita algún recibo.

Can I use recibo to ask for the bill at a restaurant?