If you want a call directly from a browser, you will probably need a WebRTC integration. This is possible with Jabber Guest, or the relatively new CMS product, which can be achieved with an ExpressWay Edge architecture.
See below for CMS:
I would recommend CMS, as this is more in line with Cisco's latest arch trends, and has native WebRTC, which doesn't require a client-side plug-in install.
Finally, the customer could go for a Spark Hybrid design, and embed spark as a widget into a webpage, and do a behind-the-scenes integration to reach your on-prem system.
However, whatever you'd choose - you're going to have to upgrade from CUCM 7.1, which has been EOL for some time. Until you do that, any discussion around this type of integration is largely academic.