How do the rules of vocal harmony apply to this sentence? In other words, why doesn't the indicator of plural accusative copy the vocal "ö" from the plural form of "mérnök"? Why is it not "öt"?
For some reason, plural nouns can only end in "-at" or "-et" in accusative.
Vocal harmony in Hungarian usually is complicated and has many exceptions. The vowel harmony in Turkish is much more structured and logical
Yep, Turkish vowel harmony is pretty simple and the exceptions are few and easy to remember. Finnish vowel harmony is also simple.
Why would "I look for engineers" be considered incorrect?
If you really need them it should be accepted. Of course, it is OK.
Why "I am looking for the engineers" was not accepted??
Simply because you're not looking for certain engineers. Just engineers in General.
The translation of your sentence would involve an additional article (like in English), and the use of definite conjugation: "Én a mérnököket keresem."
How do I say "I am looking for my socks" in Hungarian.
A zoknimat keresem.