How do americans pronounce inquiry?

Americans may pronounce it either way.  We have a very big country, a large population, and a lot of regional and ethnic differences in the pronunciations of many words.

In-qua-ree.  Some pronounce it 'in-quIre-ee,' which is basically saying the word 'inquire' with a long 'e' (ee) sound at the end,

On the first day of my freshman year, one of the professors happened to mention that "inquiry" should be pronounced "in-QUIRE-ree."

I looked it up and never pronounced it "IN-que-ree" again.

A couple of years later, one of my Bio professors told us that "abdomen" is pronounced "ab-DO-men," but I was no longer so impressionable, and I didn't care.

But I read somewhere that it doesn't make any sense to pronounce "envelope" with a French first syllable and an English last syllable."ON-vee-lope" doesn't make any sense, and even "ON-vee-lop" has no reason to be in French

According to whatever it was that I was impressionable about, it should be EN-vee-lope," and that's how I always say it.

In my experience, Americans generally say 'IN-quir-ee', with the stress on the first syllable.

British people generally say 'in- QUIRE-ee'.

While I'm aware of other pronunciations, often mispronunciations, it is pronounced in-quih-ree, accent on the first syllable.

seems like both are used

https://forvo.com/word/inquiry/#en_usa

Which dictionaries? Dictionary.com gives two possible pronunciations- one with 4 syllables and the stress on the second, and one with 3 syllables and the stress on the first.

in-kwahyuhr-ee, in-kwuh-ree

Both are acceptable in American English. 

Your request has been denied. We don't cater to the likes of you.

"ING-kwer-ree," not "in-KWY-ree."