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The Witt group $\mathcal{W}$ of braided fusion categories (see also the sequel paper) can be defined over any field; I am happy to restrict to characteristic $0$ if it matters.
Is $\mathbb k \mapsto \mathcal W(\mathbb k)$ an (affine?) algebraic group scheme?
Assuming $\mathcal W$ is sufficiently scheme-like for the following question to make sense, what I really want to know is:
What is the etale homotopy type of $\mathcal{W}$?
The answers to your questions are essentially in the first paper you cite. The second paper has more information on finer structure, like torsion, but the basic properties are all we need.
In the first paragraph of the introduction, the authors mention that any embedding of algebraically closed fields of characteristic zero yields an isomorphism on rational points, so it is geometrically a zero dimensional object. In particular, it is perhaps more profitable to think of it as a big Galois module. By Corollary 5.23, $\mathcal{W} \otimes \mathbb{Q}$ is a vector space of countably infinite dimension, so (assuming one manages to define $\mathcal{W}(R)$ for commutative $\mathbb{Q}$-algebras $R$) it is in fact an ind-affine group ind-scheme, rather than an affine group scheme.
