Hi Adam,
I use Intercom for my SaaS product (www.storemapper.co) and work with lots of merchants on Shopify. Intercom is awesome, but really shines when every person interacting with your site is actually logged in and has a user account and email associated. Intercom doesn't really have a way to communicate with a not-logged-in user (that I know of). Some of your Shopify customers will create accounts after their first purchase and then when they visit your site they will be logged in and you could pretty easily pass that user data to Intercom, but that's the only situation in which you would be able to interact with/track your users.
I agree with Aaron that something more like olark.com would be better for messages visitors to your site and mixpanel.com, kissmetrics.com or keen.io would be better for tracking user behavior, funnels and the like.
Cheers,
Tyler
