I have a site that is a few years old (https://goo.gl/vMUooK) and have been asked to make it responsive. Is it better to start the design from scratch or is there a way to easily change to convertible layout?
Refactoring for accessibility
I've taken a quick look-see at your index page. Your markup is generally good, but will require a (very) few changes in order for it to be responsive to a variety of UAs.
There are issues with the css. Its having fixed metrics, for example, causes the page to tend to break on resize. It is also user unfriendly for persons with visual difficulties.
I am going to pass the buck here. There is an old thread on another forum that carefully walks through the process of coding an accessible, responsive page. The author I'm linking is hard-headed, obnoxious and very knowledgeable.
Read and absorb his posts. With practice, you'll know how to code responsive sites that are fully accessible to all, and do it in your sleep.
When you run into something that's unclear to you, I'll be glad to help you find clarity.