Can the set of all sets be vacuously determined? Can I say that the empty set is vacuously the set of all sets since there isn’t anything in it to disprove such statement?
But in this case, I guess it would turn out to be an impossibility to vacuously determine the set of all sets since its power set would reveal to be an infinite not vacuous set: $\mathcal{P}(\emptyset)$, $\mathcal{P}(\mathcal{P}(\emptyset))$, $\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset))))$, $\dots$
 A: Two issues: (i) In standard set theory, one can prove that there is no such thing as “the set of all sets”. And (ii) You are misinterpreting what “vacuous truth” means.
To address the latter first: the notion of “vacuous truth” comes from the fact that the negation of a statement of the form “For all $x$ ($P(x)$ is true)” is the statement “There exists an $x$ such that ($P(x)$ is false)”.
As a consequence of this, given any property/statement $P$, you can affirm that “For all $x$ in the empty set ($P(x)$ is true.” Because the negation of this statement would be “There exists an $x$ in the empty set for which $P(x)$ is false”. And in order for that latter statement to hold, there has to be an element in the empty set, which is impossible (the fact that it’s supposed to make $P(x)$ false doesn’t even matter; you can’t get started). We refer to this phenomenon by saying that “for all $x$ in the empty set, $P(x)$ is true” is vacuously true. So, “every $x$ in the empty set is green” is true; so is “every $x$ in the empty set is red.” So is “every SUV that I currently own is peach” (I don’t own any SUVs at the moment). All of these are “vacuously true”. Note that all of them are of the form “For all $x$ in the empty set, (something).$
Now, $X$ is the “set of all sets” if and only if the following is true: “for every $x$ that is a set ($x\in X$).” Note that it is not a statement of the form given above. So you ask: is the empty set the set of all sets by vacuity? No: because $\varnothing$ is a set, yet $\varnothing\notin \varnothing$. So you cannot substitute $\varnothing$ for $X$ in the statement above and get something that is true. Note that we are not quantifying over all elements of the empty set (there is no “for all $x$ in the empty set” statement). So “vacuity” does not come into play at all.
Note also that this is regardless of whether there is such a thing as a “set of all sets”. Whether there is or there isn’t one, it is clear that the empty set cannot be it.

Now, separately one can prove that in fact there can be no set of all sets in standard set theory. There are multiple ways of doing it; one common one is to establish a variant of Russell’s paradox (alluded to by @Bernard in the comments), and prove that:
Theorem. In Zermelo-Fraenkel Set Theory, for every set $A$, there is a set $B$ such that $B\notin A$.
(In fact, similar arguments hold in other axiomatizations of set theory)
Proof. Let $A$ be a set. By the Axiom of Separation, the collection of all elements of $A$ that satisfy a given condition is itself a set. So let
$$B = \{x\in A\mid x\notin x\}.$$
So $B$ is a set. I claim that $B\notin A$.
Indeed, assume to the contrary that $B\in A$. Then either $B\notin B$ or $B\in B$. If $B\notin B$, then since $B\in A$ and it satisfies the defining condition, we conclude that $B\in B$. But if $P$ implies not($P$), then $P$ is false. So $B\in B$.
But if $B\in B$, then $B$ does not satisfy the defining condition, so $B\notin B$. Thus we conclude that $B\notin B$.
Hence $B\in B$ and $B\notin B$. This is a contradiction. It arises from the undischarged assumption that $B\in A$, so we conclude that in fact $B\notin A$.
Thus we have shown that given any set $A$, there is always a set $B$ that is not an element of $A$. $\Box$
Corollary. In Zermelo-Fraenkel Set Theory, there is no “set of all sets”.
Proof. We just proved that for every set $X$, there exists a set $x$ such that $x\notin X$. So no set can be “the set of all sets.” $\Box$

Note that the second part has nothing to do with “vacuous truths”. And the first part has nothing to do with “paradoxes”.
A: What is vacuously true is that every element of the empty set is a set (or more generally, any property is true of every element of the empty set).  That is, for all $x$, if $x\in\emptyset$ then $x$ is a set.  To say the empty set is the set of all sets, however, would also require the converse: that for all $x$, if $x$ is a set, then $x\in\emptyset$.  This is not true: there are lots of sets that aren't elements of the empty set (indeed, every set is a counterexample).
