Need a process to get current clients to give us personal introductions to others
You've received some good help here, already, so I won't repeat all of that. Instead, I'll highlight one of the very simple but crucial things you can do, and that's to help the client you are depending on tell your story in the right way. The longer they have been your client, the less likely they will introduce you properly. That's because they may have you in a box that you've long ago busted out of. I think you know what I mean, too.
Picture a lead coming in by email, and you set up a phone call to explore the relationship. Somewhere in the call you'll ask, "Hey, how did you hear about us?" What they say next will make your heart sink or soar, because you'll immediately know--based on who introduced you--what impressions they already have. You're quick, or cheap, or fun to work with, or a great problem solver, etc.
So don't take that for granted--it's not just getting them to introduce you, but feeding them the right story to tell so that you're far down the road of being positioned appropriately.