Let $V$ be a (Hausdorff) topological vector space, and let $S = \{v_i\} _{i \in \mathbb N} \subset V$ be a generating subset (notice that it is countable!).
Does countability make it possible to prove the existence of a Schauder basis $B \subseteq S$ (contained in $S$) without Zorn's lemma?
With or without Zorn's lemma, one cannot construct a Schauder basis without detailed knowledge of the space. It is not a thing that drops out from some abstract argument. There are topological vector spaces, even Banach space, with no Schauder basis at all.
Even if the space $V$ has a Schauder basis, we shouldn't expect to obtain it from some set of vectors with dense linear span. Being a Schauder basis requires much more than being a linearly independent set with dense linear span. One has to be able to represent every vector in the space as the limit of partial sums of a series $\sum c_n x_n$, not just by some arbitrary sequence of linear combinations of $c_n$.
A standard example: The set of monomials $S = \{x^n : n=0,1,2,\dots\}$ has dense linear span in $C[-1,1]$ (the space of continuous functions on $[-1,1]$ with uniform norm), because every $f\in C[-1,1]$ can be uniformly approximated by polynomials (Weierstrass' theorem). It is also linearly independent, since the only polynomial that is identically $0$ is the polynomial with zero coefficients. But $S$ is not a Schauder basis, because the continuous function $f(x) = |x|$ does not admit a representation $f(x) = \sum_{n=0}^\infty c_n x^n$. Indeed, the functions with a power series representation are infinitely differentiable on $(-1,1)$, where one can differentiate the power series term by term.
The above example also answers your question in the negative. No subset of $S$ is a Schauder basis, since throwing out some monomials will not help us represent the function $f(x)=|x|$ as a series.