In movie credits, you will see wranglers of animals, such as horse wrangler, ant wrangler (Pi), tiger wrangler, etc.
However, you will notice other things are wrangled, including children, automobiles, and puppets.
I suspect that the term wrangler originated in Hollywood westerns, when horse wranglers were called in to handle horses, and then the term was extended to other people and things on the set.
In my googling, the earliest reference I found was for puppet wrangler in Follow That Bird in 1985.
When is the earliest use of "wrangler" for something other than an animal?