Conventionally, we interpret a first-order theory by assigning extensions (sets of objects) to its predicates, and fixing a domain (another set of objects) for its quantifiers to run over. And then we appeal to some modest set-theoretic ideas to do the metatheory for our first-order theories.
But it is a nice question whether we need to invoke sets here. We might wonder: why not directly assign a (monadic) predicate the objects, plural, which satisfy the predicate, rather than something over and above those objects, namely the set of them? Why not treat quantifiers as ranging over some objects, plural, rather than something over and above them, namely the set of them?
Well, to handle a metatheory which deals in objects, plural, we'll need a plural logic. See for example Plural Logic by Alex Oliver and Timothy Smiley: in Ch. 11 they give an account of the semantics of standard first-order logic using a plural metatheory with no commitment to sets. So this can be done.
It is, however, an interesting question what exactly is gained by the exercise. OK we'll avoid ontological commitment to sets. But we'll still find ourselves tangling with some strongly infinitary ideas. For Oliver and Smiley have to trade in arbitrary sets (whose membership we can't specify) for an unreduced notion of arbitrary properties and arbitrary functions (whose graphs we can't specify); and someone who is worried about the notion of an arbitrary infinite set will argue that we should be equally worried about these substitutes. But that's a debate for another day.