Learn Turkish: Komşular davetiyeyi almış mı? - Have the neighbors supposedly received the invitation?

The suffix -mış is the inferential or reported past tense. If you used aldı mı, you would be asking a direct factual question: Did they receive it? By using almış mı, you are asking about something you haven't witnessed directly, adding a sense of hearsay or seeking confirmation: Have they supposedly received it?

Why does davetiye have -yi at the end?

Because it is a specific direct object, meaning the invitation, so it needs the accusative case. Since davetiye ends in a vowel, we add the buffer letter y followed by the suffix -i to make davetiyeyi.

Since komşular is plural, shouldn't the verb be almışlar mı?

You can say almışlar mı, but it is not required. In Turkish, when a human plural subject like komşular is explicitly stated in the sentence, the plural suffix on the verb is usually dropped to avoid repetition and sound more natural.

Doesn't almak usually mean to buy or to take?

It does! However, when the object is something sent or given to someone, like an invitation (davetiye), a letter, or a package, the verb almak translates naturally to receive or get in English.