Yes, there is.
(As said in the comments) the function $f=\emptyset$ is the bijection you seek.
It may seem strange that $f$ is a set, but this is ok since the definition of a function is a set of pairs (s.t ...), in this case we take $f$ to be an empty set.
The definition of $f$ is fine since there are no $x, y_1,y_2 \in \emptyset$ s.t $y_1\neq y_2$ and s.t $(x,y_1),(x,y_2)\in f$.
$f$ is injective since there are no two different elements in $\emptyset$ that $f$ maps to the the same element (this is just because there are no two different elements in $\emptyset$).
$f$ is surjective since for each $y$ in $\emptyset$ there exist a source (this is since there are no elements in $\emptyset$ hence we can't say there exist an element that doesn't have a source).
The empty set is indeed a set (the set of no elements) and it is a subset of every set, including itself. $$\forall A: \emptyset \subseteq A,\;\text{ including if}\;\; A =\emptyset: \;\emptyset \subseteq \emptyset$$
$$\text{BUT:}\quad\emptyset \notin \emptyset \;\text{ (since the empty set, by definition, has no elements!)}$$
That is, being a subset of a set is NOT the same as being an element of a set: $$\quad\subseteq\;\, \neq \;\,\in: \;\; (\emptyset \subseteq \emptyset), \;\;(\emptyset \notin \emptyset).$$
$\emptyset \;\subseteq \;\{1, 2, 3, 4, 5\},\quad$ whereas $\;\;\emptyset \;\notin \;\{1, 2, 3, 4, 5\},\;$.
$\{3\} \subseteq \{1, 2, 3, 4, 5\},\quad$ whereas $\;\;3 \nsubseteq \{1, 2, 3, 4, 5\}, \text{... but}\; 3 \in \{1, 2, 3, 4, 5\}$.
Best Answer
Yes, you are right. It is vacuously true. Here's a more detailed explanation of why:
In math, either a statement is true, or its negation is true (but not both). That means, for example, either the statement (a) $\forall x \in \emptyset$, $x^{2} = 1$ or its negation, (b) $\exists x \in \emptyset$ such that $x^{2} \neq 1$, is true, and the other is false.
It's clear that statement (b) is false since $\exists x \in \emptyset$ is a false statement. So, since statement (b) is false, its negation, statement (a), must be true (it's called vacuously true).