You can record the response rate by using the inbuilt Zendesk method for native CSAT surveys.
Note that CSAT surveys are only available on the Professional plan or higher so this method will not be possible on other plans.
The way this method works is that in the trigger that sends the survey, under "Meet all of the following conditions" set the rule "Ticket: satisfaction" - "is" - "unoffered" like this:
Then under the "Actions" section of the trigger, add the rule "Ticket: satisfaction" - "Offered to requester" like this:
Great, you're done.
The reports showing you your response rate already exist over in Explore on your "Zendesk Support" dashboard, under the "Satisfaction" tab like this:
You'll then need to scroll to the bottom to see the following report: