I know that the cosmological principle states that the universe is both isotropic and homogeneous, but I was wondering if these both had to exist at the same time?

No, they don't necessarily have to, in general.

For example, a uniform electric field in the $\hat x$ direction is a homogeneous vector field. But it is not isotropic.

Such an electric field is homogeneous because it obeys: $$ \vec E(\vec r) = \hat x E(\vec r) = \hat x E(\vec r + \vec d) = \vec E(\vec r + \vec d)\;, $$ for any $\vec d$.

But an active rotation, say by $\pi/2$ about the $\hat z$ axis, changes the components of $\vec E$. $$ R(\pi/2;\hat z)\vec E(\vec r) = E(\vec r)R(\pi/2;\hat z)\hat x = E(\vec r)\hat y \neq \vec E(\vec r)\;. $$

Is there a way that the universe could be isotropic, but not homogenous or the other way around and what would that look like?

Yes, things can be isotropic but not homogeneous.

For example, suppose you are considering isotropy with respect to the origin, $\vec r_0 = (0,0,0)$, of some coordinate system.

The any function $f(\vec r)$ that only depends on the magnitude of $\vec r$ ($r$) is isotropic, but not homogeneous (because points with different $r$ values can have different $f(r)$ values).

I know that the cosmological principle states that the universe is both isotropic and homogeneous, but I was wondering if these both had to exist at the same time?

No, they don't necessarily have to, in general.

For example, a uniform electric field in the $\hat x$ direction is a homogeneous vector field. But it is not isotropic.

Such an electric field is homogeneous because it obeys: $$ \vec E(\vec r) = \hat x E(\vec r) = \hat x E(\vec r + \vec d) = \vec E(\vec r + \vec d)\;, $$ for any $\vec d$.

But an active rotation, say by $\pi/2$ about the $\hat z$ axis, changes the components of $\vec E$. $$ R(\pi/2;\hat z)\vec E(\vec r) = E(\vec r)R(\pi/2;\hat z)\hat x = E(\vec r)\hat y \neq \vec E(\vec r)\;. $$

Is there a way that the universe could be isotropic, but not homogenous or the other way around and what would that look like?

Yes, things can be isotropic but not homogeneous.

For example, suppose you are considering isotropy with respect to the origin, $\vec r_0 = (0,0,0)$, of some coordinate system.

Then any function $f(\vec r)$ that only depends non-trivially on the magnitude of $\vec r$ is isotropic, but not homogeneous (because points with different $r$ values generally have different $f(r)$ values).