Is there a "canonical" way to define open/closed/constructible subsets of a locally free sheaf/vector bundle $\mathcal{F}$ on a scheme $X$? There is a clear way to do this with topological vector bundles, but are things defined the same way in algebraic geometry?
I came across them in a paper (https://arxiv.org/abs/1910.05207), but couldn't find any other sources discussing possibly related concepts specifically in algebraic geomery other than those looking at subsets of global sections of these vector bundles.
My guess was that this has something to do with interpreting a locally free sheaf/vector bundle $\mathcal{F}$ locally as (relative) $\text{Spec} (\text{Sym}^\cdot \mathcal{F}^\vee)$ (which are also discussed in section 4 of the paper above), but I'm not sure how to formulate a precise definition other than what was suggested at the beginning of this question. It may be helpful to list interesting examples/non-examples.
Your guess is correct. The construction, mentioned in section 4 of your link, is that given a scheme $X$ and a locally free sheaf $\mathcal{F}$, we can define the geometric vector bundle associated to $\mathcal{F}$ as the relative spectrum $\underline{\operatorname{Spec}}_X(\operatorname{Sym}^\bullet(\mathcal{F}^\wedge))$. On an open set $U\subset X$ where $\mathcal{F}$ is trivial (that is, $\mathcal{F}|_U\cong \mathcal{O}_X^n|_U$) with basis given by the sections $s_1,\cdots,s_n$, this construction gives that our geometric vector bundle is exactly $U\times \Bbb A^n$ where the $s_i$ are exactly the "basis directions" of the $\Bbb A^n$ factor. This is the same as what one would expect from the topological side of things: on a trivializing open set $U$, we can write a vector bundle as the product $U\times \Bbb R^n$.