Let $X$ be a variety (i.e. a reduced scheme of finite type over a field) and let $G$ be an abstract group, finitely generated, acting of $X$ algebraically freely. The example I have in mind is $\mathbb Z$ acting by shifts on $\mathbb A^1$. The quotient in this example clearly does not exist as a noetherian scheme since the fibres are discrete infinite. Is it possible to still make sense of the quotient algebraically? I guess the best way to put it formally is:

Is it possible to put the category of varieties over the field k into a bigger "algebraic" category such that the functor of points of the quotient (i.e. orbits of the values of the usual functior of points) is always representable? What I mean by "algebraic" is that I am definitely not interested in a quotient in the sense of complex geometry.

Is it possible to achieve this goal by considering a suitable category of non-Noetherian schemes?