Is "Whom should I give this job to?" grammatically correct?

In modern colloquial English, "who" is always okay. In your example, you have correctly applied the rule for old-fashioned and formal English -- it would be "Whom should I give the job to?", or perhaps better (in that style): "To whom should I give the job?" (But "To who should I give the job?" sounds wrong.)

Greg pointed the right solution : By using to whom, you must put "whom". In US colloquial speech we use who for whom & it's fine. But here, "To whom" sounds the best version.

Plus, this "Whom" is object to receive the job, so as it is not active, it should not be "Who" whatever we may say today.

If you choose the tag "whom" you'll find about 50 posts about the topic whom or who. "whom" is the normal accusative but language is changing and in spoken language the m of whom is dropped due to the fact that in English nominative and accusative have the same form with the exception of a handful of personal pronouns (special accusatives: me him her us them). And in written language whom is becoming formal.