The sentence "I will await your response" is correct and usable in written English. You can use it when you are asking someone to reply to you or when you are expecting to receive a response from someone. For example, "I have sent you my project proposal. I will await your response.".
I will await your success story".
I eagerly await your response.
Dear Texas Department of Public Safety, thank you for taking the time to read this, and I eagerly await your response.
I call for creation of a Denver Art Caucus, and I await your response.
"I await your response".
"I await your response, alongside many Mexicans," he finishes.
"We will await your Captain Gordon in the drawing-room".
Being a terminologist, I care about word choice. Ludwig simply helps me pick the best words for any translation. Five stars!
Maria Pia Montoro
Terminologist and Q/A Analyst @ Translation Centre for the Bodies of the European Union