Learn Turkish: Göndermeleri icap eden tebliğ postanede. - The notification that they are required to send is at the post office.

It literally translates to "which necessitates their sending." First, we take the verbal noun gönderme (sending) and add the possessive -leri (their) to make göndermeleri. Then we add icap eden, which is the active participle (the -An form) of icap etmek (to necessitate / to require). Placed before the noun tebliğ, it forms a relative clause of necessity: "The notification that necessitates their sending."

Could göndermeleri also mean "his/her sending them"?

Yes, grammatically it is ambiguous out of context. It can be broken down as gönderme + -leri (their sending) OR as gönderme + -ler (plural marker for the objects being sent) + -i (his/her sending). Here, the English translation clarifies that the subject is "they," so it means "their sending."

Could I use gereken instead of icap eden in this sentence?

Yes, göndermeleri gereken tebliğ is perfectly correct and means the exact same thing. İcap etmek (to be necessary) is just a slightly more formal and older synonym for gerekmek. It is used exactly the same way to form these clauses.

Is tebliğ the standard word for any kind of notification?