Turn "we're happy with our current vendor" into a discovery conversation instead of a conversation-ender.
When a prospect says "We're already using [Competitor]" or "We're happy with what we have," respond: "That's great — most of our best customers said the same thing before they switched. I'm curious what you like most about them?" The question is genuine and surfaces the specific value they place in the current solution. From there, you either qualify out (they've solved the problem) or discover where the current solution falls short — which is your opening.