"Inquiry" - IN-kwer-ee or in-KWIRE-ee?

in-KWIRE-ree

I'm a native American English-speaker, and I've never heard the first pronunciation..

Either or both, depending on your choice.  I tend to use in kwer ee, myself.  I do sometimes use in kwire ee though.  Usually when I emphasize the word as a somewhat formal term, rather than pass over it as just another part of the sentence.

The inkwireee lasted two hours, that was a long time for a basic inkweree.

IN- kwer- ee. This is the most common in American English. At least on the west coast.

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Both pronunciations are proper. Which one one would use depends on one's accent or dialect. American accents tend towards "INkwerree" and "INGkwerree," while British accents tend towards "inKWYree."