Wow! You have a good sense of English here. That sentence is redundant and a little bit unnatural--although a native speaker might say it in some context.
If you wanted an expression that keeps both parts of the sentence, it could be something more like this:
"We will investigate your inquiry as quickly as possible and then get back to you."
Or, instead of investigate, you could use "look in to."
NOTE: "Inquiry" and "enquiry" are both used in American English. But in British English, "enquiry" may be more common.