Learn Turkish: Bu semineri destekleyecek sponsorları bulalım mı? - Shall we find the sponsors who will support this seminar?

This is the future subject participle. It is formed by adding -(y)ecek to the verb stem destekle- (to support). The buffer -y- is used because the stem ends in a vowel. Because it acts like an adjective describing the subject of the action (sponsorları), it does not take any personal endings.

Why do both semineri and sponsorları have accusative case endings?

They are specific direct objects for two different verbs! Semineri is the object of the participle destekleyecek (to support this seminar). Sponsorları is the object of the main verb bulalım (shall we find the sponsors). Participles still act like normal verbs for the words placed immediately before them.

I know bulalım means "let's find", so why does the translation say "shall we find"?

When you take a 1st-person plural "let's" form (using the optative ending -alım or -elim) and add the question particle mı, it turns a proposal into a question. You are asking for the listener's opinion, so "let's find" naturally translates to "shall we find?" or "should we find?".

Why does the phrase bu semineri come at the very beginning of the sentence?