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.

Agree with your analysis and Megan's answer. To make the sentence *not* redundant, you can change "respond to" to "address." --> We will address your inquiry as quickly as possible and then get back to you. (or, we will address your inquiry and respond to you as soon as possible).

Well it is a tad redundant but it does make the speakers point clear.