I think we can't construct, for example, the set of all vector spaces, because we can't construct the set of all sets.
However, there are theorems like "every vector space has a basis", which is a universal quantification of "$x$ has a basis" over the set of all vector spaces.
How do we quantify over structured sets?
Set theory is formalized in first-order logic, where quantifiers range over the whole domain of discourse. For set theory, the domain of discourse is the universe of all sets, so every "for all" sentence is (in a sense) about all sets.
To "quantify over less", we use conditional statements: as pointed out in the comments, we can write "if $x$ is a vector space, then $x$ has a basis" as a first order formula $\varphi$ with free variable $x$, so $\forall x \varphi$ is a first-order sentence stating that every vector space has a basis. In this way, we can effectively quantify over any class definable by a formula.