I'm currently advising a client right on exactly that kind of customer service integration.
Look into Live Ninja (https://liveninja.com/). It brings together the best of both worlds from web and a call center. Voice + Video + Instant connect without any special software, or a phone. You can have either one way, or two way video, and text chat too.
Obviously, get nice uniforms for your call center guys if you haven't already.