Defining first-order logic doesn't really require set theory, but it does require some conception of natural numbers. Instead of saying there's an infinite set of variables, you can have a single symbol x and a mark *, and then you can say a variable is any string consisting of x followed by any number of marks. You can do the same thing with constants, relation symbols and function symbols. This does mean there can only be countably many of each type of symbol but for foundational purposes that's fine since each proof will only mention finitely many variables.
Allowing uncountably many symbols can be more convenient when you apply logic in other ways, e.g. when doing model theory, but from a foundational perspective when you're doing stuff like that you're not using the "base" logic but rather using the formalized version of logic that you can define within the set theory that you defined using the base logic.
Allowing uncountably many symbols can be more convenient when you apply logic in other ways, e.g. when doing model theory, but from a foundational perspective when you're doing stuff like that you're not using the "base" logic but rather using the formalized version of logic that you can define within the set theory that you defined using the base logic.