Garter snakes do need amphibians as part of their diet. Garters on a mouse diet do not suffer from nutrient deficiency, but not every garters will accept pinkies. The most common one of these deficiency are: Vitamin B (frozen fish, goldfish), vitamin E and calcium (fish does not provide enough, worms don't provide any at all).
Amphibians make about 70% of an adult diet, with fish and worms being the other 30% according to Schmidt and Davis.
A friend of mine have a garter and he bred his own frogs until the snake was big enough to accept a pinkie.
However there is a fix for all this -- vitamin and calcium suppliments on frozen fish.