In the foundations I'm familiar with, you don't ever "create" anything.
Take ZFC for example. If we need the real numbers, we search in our static universe for a set $\mathbb{R}$ of the correct cardinality such that its not-too-difficult to define some other sets, namely
$$+,\cdot,0,1,\leq$$
such that its not too difficult to show that $\mathcal{R} = (\mathbb{R},+,\cdot,0,1,\leq)$ satisfies the axioms of a complete ordered field. So in some sense, we're "searching" in a static universe, as opposed to "creating" in a dynamic universe. The only things that are changing are our definitions.
Are there any foundations in which the universe itself gets dynamically extended?
I'm not certain how this would work exactly, but perhaps something like this. Instead of having axioms postulating the existence of different entities (an empty set, a powerset etc.), we might instead have methods for constructing new entities (a new empty set, a new powerset, etc.).
Edit. Here's some more details about what I'm looking for.
Consider the following statement.
There exists $x$ such that for all $y$ we have $x \neq y$.
Clearly, its bogus. Now consider the following variant.
We can always construct a new entity $x$ such that for all previously existing $y$ we have $x \neq y$.
It would be nice if the above statement were true.
More ambitiously:
For all models $\mathcal{R}$ over a signature $\sigma$, we can construct a new (abstract) model $\mathcal{R}'$ over $\sigma$ equipped with an isomorphism $f : \mathcal{R}' \rightarrow \mathcal{R}$.
This is exactly what happens in the constructive foundations that are based on a computational view of mathematics, for example Martin-Löf's type theory (both the extensional and the intensional one).
In these foundations, mathematical entities must be computable, so every set must be inductively generated. This means that until we don't say how the elements of a certain set are formed, it doesn't make sense to say that the set itself "exists".
I'll give you an example to explain why (or how) type theory can satisfy your request. I won't use Martin-Löf's type theory because in that system the rules are very long since they deal with equality in detail; what follows is based on (a trivial extension of) simply typed $\lambda$-calculus. I hope you're familiar with Gentzen-style sequent calculus: if you're not, I'll explain the notation.
Suppose that your initial universe contains only one basic set, namely, a set $B$ with two elements. The only other construction available is that of function sets, because there's nothing you can actually do without it since all computations come from abstraction and application. This means that your universe is presented through the following list of rules:
$B$-formation: this rule declares that we can talk about $B$ as a set. $$\vdash B \;\text{set} $$ $B$-introduction: this rule tells how canonical elements of $B$ are formed. $$\vdash \mathsf{0} \in B \quad \vdash \mathsf{1} \in B$$ $B$-elimination: this rule tells what information we can get from knowing that something is an element of $B$. $$\frac{\Gamma \vdash c \in B \qquad \Gamma \vdash d \in \alpha \qquad \Gamma \vdash e \in \alpha}{\Gamma \vdash \mathsf{R}_B (c, d, e) \in \alpha}$$ with the reduction rules that tell us what to do when $c$ is a canonical element: $$\mathsf{R}_B (\mathsf{0}, d, e) \rightsquigarrow d \qquad \mathsf{R}_B (\mathsf{1}, d, e) \rightsquigarrow e$$
Now, we could start enumerating all the different sets within our universe. They are: $$B,\quad B \to B,\quad B \to (B \to B),\quad (B \to B) \to B, \quad \dotsc$$
The notion of element equality, in this system, is supposed to correspond to the smallest equivalence relation containing reduction. The notion of set equality, instead, is purely syntactical. We don't have any extensionality rule here. This means that we can add to our list of rules some new rules for a set $C$, for example a set with three elements, and it will be different from $B$ – but even if the rules for $C$ had exactly the same form as those of $B$, in a way that would make $C$ another set with two elements, the constructors would be different. We could call them $\mathsf{0}'$ and $\mathsf{1}'$, or $\mathsf{a}$ and $\mathsf{b}$. These are all just symbols, they're not to be intended as variables: the fact that $B$ and $C$ are different comes from the fact that B and C are different letters!
I hope this clarifies my initial answer.