I don't think I have ever seen a 16" geared bike, not to say that they do not or have not existed.
I could see the tolerances getting tight with a geared bike that size. Our 20" geared bike has the derailleur getting close to the rim. That could be fixed with a shorter cage der. though. The chain line on the smaller wheel would cause greater chain angles. The chain might complain some at the extremes (1st and 7th gear). You could build your own from parts. The cost would likely be near the cost of a new bike but it could be a fun project.