Research "multi-tenant database architecture". (For your purposes, think of one tenant as one client.) You'll find a spectrum from "one database per tenant" to "every tenant in every table".
Read carefully. Writers in this field can confuse you. Expect technical terms like shared schema to mean different things to different writers.
See this SO answer for tradeoffs.
For your first step, I wouldn't consider anything besides simply moving those client databases in-house. Just doing that is going to give you and your application programmers enough headaches. You don't need an architectural change on top of it.
That will also give you time for research and testing.