Learn Turkish: Müşterileri bekliyor. - Their customer is waiting.

The suffix -leri here is the third-person plural possessive meaning "their", attached directly to the singular word müşteri (customer). Because the owner is plural (they), but the item is singular (customer), we get "their customer".

Could "müşterileri" also mean "his customers" or "her customers"?

Yes, it absolutely could! Turkish words can sometimes be broken down in multiple ways. It could be müşteri + -leri (their customer) OR müşteri + -ler + -i (his/her customers). Without a larger story around the sentence, context is the only way to tell them apart.

Why doesn't the sentence start with "onların" (their)?

You could choose to say Onların müşterileri bekliyor, but because the suffix -leri already clearly tells us the customer belongs to "them", the word onların is usually dropped. You only keep it if you really want to emphasize whose customer it is.

I learned the verb "to wait" is "beklemek". Why is it spelled "bekliyor" instead of "bekleyor"?

When a verb root ends in the wide vowels -a or -e (like bekle-), that vowel changes to fit the present continuous suffix -iyor. The e narrows to an i to match the sound rules, giving us bekliyor.