Topluluk üyeleri, deprem sonrası kurtarma çalışmaları için gönüllü oldu.

In Turkish noun–noun compounds, the first noun stands alone (no genitive ending), but the second noun takes the 3rd-person possessive suffix to show “of.” So:
• üye = member
• üye + ler = üyeler (members, plural)
• üyeler + i (3rd-person poss.) = üyeleri (its/their members)
Thus topluluk üyeleri literally means “the community’s members” but is freely used as “community members.”

• deprem sonrası is a noun-noun compound (“post-earthquake”) where sonrası = sonra (after) + 3rd-person poss. -sı (“its after”). It functions like an adjective: “post-earthquake.”
• depremden sonra uses the ablative -den + sonra to form an adverbial phrase: “after the earthquake.”
You can use both, but deprem sonrası is more nominal/attributive, while depremden sonra is purely temporal.

Turkish allows a singular verb with a plural subject when you treat the group as a single unit or focus on the collective action.
• Topluluk üyeleri gönüllü oldu. (They volunteered as a group.)
If you want to emphasize individual actions, you can use the plural: oldular.

gönüllü is an adjective (or noun) meaning “volunteer” or “willing.” To say someone “became a volunteer” or “volunteered,” Turkish uses the copular verb olmak (“to become/to be”).
• gönüllü + oldu = “(they) became volunteers.”

It’s a compound of two nouns:
• kurtarma = rescue (from the verb kurtarmak)
• çalışma = work, effort, operation (from çalışmak)
Together, kurtarma çalışması means “rescue operation,” and kurtarma çalışmaları is the plural: “rescue operations” or “rescue efforts.”

için is a postposition meaning “for” (indicating purpose or benefit). It attaches directly to a noun phrase without any additional case ending.
• kurtarma çalışmaları için = “for rescue operations.”

Turkish generally follows S-(Time/Manner/Purpose)-Verb. In this sentence:
S = Topluluk üyeleri,
Time = deprem sonrası
Purpose = kurtarma çalışmaları için
Verb = gönüllü oldu