Let me work with arbitrary affine schemes $X = \text{Spec } k$. I believe there is a reasonable notion of a spin structure on a quadratic module $(V, q)$ over $k$, by which I mean a pair consisting of a finitely generated projective $k$-module (vector bundle over $X$) and a map $q : V \to k$ such that
- $q(\lambda v) = \lambda^2 q(v)$ for all $v \in V, \lambda \in k$, and
- $b(v, w) = q(v + w) - q(v) - q(w)$ is bilinear.
Namely, any such thing has an associated Clifford algebra $\text{Cl}(V, q)$, and a reasonable notion of spin structure in this context is that a spin structure on $(V, q)$ is a super Morita trivialization of $\text{Cl}(V, q)$, or somewhat more explicitly a finitely generated projective super $k$-module $M$ such that $\text{End}_k(M) \cong \text{Cl}(V, q)$ (here End is the super End) and such that $\text{Hom}_k(M, -)$ is faithful (or something like that).
In differential geometry the miracle is that any smooth manifold $M$ has a more-or-less canonically associated quadratic module (over $C^{\infty}(M)$), namely the tangent bundle of $M$ equipped with some Riemannian metric. The sense in which this is more-or-less canonical is that the space of Riemannian metrics on $M$ is contractible. But I don't see any analogue of this construction in algebraic geometry.