There are many approaches to studying any mathematical theory - ZFC or otherwise.
We can avoid models altogether and just look at syntax, formulas, and proofs. This can often be done in very weak metatheories whose only objects are natural numbers.
We can study countable models of a countable theory in metatheories like second-order arithmetic. This includes countable models of ZFC. The objects of the model are coded as natural numbers and the relations on the model are coded as sets of numbers. Second order arithmetic can define the satisfaction/truth relation on a countable model and handle many other constructions in basic model theory. We know that every model whatsoever of a countable theory has an elementarily equivalent countable model, so in a number of situations there is not a loss of generality in looking at just countable models.
We can also study models of any theory using ZFC or some other strong foundational metatheory. In ZFC, we can form the satisfaction/truth relation on any set model of any theory, including set models of ZFC. And we can perform an enormous number of model-theoretic constructions.
There's nothing contradictory about using ZFC as a metatheory to study models of ZFC, and no reason that we absolutely need to define models of ZFC differently than we define models of any other first-order theory.
In this sense, the question is a little misguided. There's no real circularity in using ZFC to study models of ZFC. And we would have the same issue studying the semantics of any other strong foundational theory like type theory within that same foundational theory.
Of course, ZFC does not prove there is any model of ZFC - this is an immediate corollary of the incompletness theorem and the consistency of ZFC. So we have to assume some amount of consistency of ZFC in the metatheory to have any models of ZFC in the first place. The same is true of any other strong foundational theory that the incompletness theorem applies to.
Here's a fable. A modern steel mill is an advanced heavy industrial complex that can produce and melt steel, adjust the chemistry, cast metal, and roll or extrude many finished steel products.
To do this, the steel mill has many complex pieces of heavy equipment. Much of this equipment is made of steel. In principle, some of the equipment could even be made of steel that was produced at the same mill.
There is no contradiction in using steel equipment to produce new steel, or in using steel from a mill to fabricate equipment for that same mill.
Of course, in the past, there were no modern steel mills. We had a variety of more primitive methods and equipment for making metal products. But we would not go back to those old methods now. We just produce steel at modern mills for whatever purpose we need it for, including equipment for steel mills.
