I know an electric field can exist without a magnetic field as in the case where you have a stationary point charge.
But, magnetic fields are created by moving charges so wouldn't you always need an electric field to have a magnetic field? Even in the case of permanent magnets, from what I know, it's the aligned moving electrons in the atoms of the material which cause the magnetic properties so doesn't that mean there's always an electric field in order to have a magnetic field?
The "magnetic field" is a concept within classical electrodynamics. Maxwell's equations were developed in the mid 19th century at a time where basic atomic physics was still a nascent field of study.
Viewed in the contemporary historical context, a permanent magnet is a perfectly fine example of a magnetic field without an electric field. Within the theory of classical electrodynamics, there is no explanation for why the magnetic field exists, only that it does exist, and how it's related to the electric field. Permanent magnets have a magnetic field as an intrinsic, fundamental property, similar to the reasons rocks have mass. They just do.
In the past one and a half centuries other theories have been developed. For example the magnetic field can be explained by special relativity as length contraction apparently creating a charge imbalance, so it could be said the magnetic field doesn't exist as a fundamental property but is rather a manifestation of the electric field in moving reference frame, and quantum physics explains permanent magnets as moving charges at sub-atomic scales.
So viewed in the context of modern physics, there's really no need for a fundamental magnetic field at all since it can be explained in terms of the electric field and motion.
The discovery of a magnetic monopole would change this, but although it would bring an elegant symmetry to the kinds of particles that exist, no evidence of a magnetic monopole has been found by experiment yet.
I suppose this is a variant of Quantum spaghettification's answer, but an obvious example is a current loop, as used in electromagnets since humans first discovered electricity.
There is no net electric field because there are equal numbers of positive and negative charges so their fields balance out. However there is a magnetic dipole due to the motion of the electrons.
No you can have a magnetic field without an electric field. Consider a rod with an equal number of positive and negative charges (such that they are equally spaced). Let the positive move to the left with speed $v$ and the negative to the right with speed $v$. This will result in a magnetic field but no electric field.
In one sense, it is an easy question, as others have pointed out. It is fairly simple to construct examples of cases with zero electric field and non-zero magnetic field.
In another sense, it's not a trivial question to answer. For example, if you see only a magnetic field in one frame, then you will see magnetic and electric field in another frame that is shifted by a change in velocity. Then there is the example of the Ahronov-Bohm effect. In this case, you have a region where both the electric and magnetic fields are zero, but an electron still feels an electromagnetic force.
The fundamental thing is the four-vector potential $A_\mu$. The electric and magnetic fields are particular arrangements of particular derivatives of this field. It is $A_\mu$ that appears in equations governing electromagnetism, such as Maxwell's equation or Dirac's equation. In several important special cases we can ignore $A_\mu$ and work with the $E_i$ and $B_i$ fields. But the fundamental understanding is always going to be based in $A_\mu$.
At a fundamental elementary particle level, the answer is that as long as no magnetic monopoles are detected, then a magnetic field dipole needs a charged particle.
The electron has a magnetic moment:
In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron caused by its intrinsic properties of spin and electric charge.
There are only limits for a magnetic moment of the neutrino, a neutral particle with a very small mass. Have a look at my answer here for further links for neutrinos.
In special relativity you can show that the following are invariant quantities, i.e. they are true in all frames: $$c ^2 \mathbf{B}^2-\mathbf{E}^2, \mathbf{B}\cdot\mathbf{E}.$$
It follows that, if one frame you have non-zero electric and magnetic fields that are perpendicular (so that $\mathbf{B}\cdot\mathbf{E} = 0$) such that $c^2 \mathbf{B}^2-\mathbf{E}^2 > 0$, then it is possible to go to a frame where the electric field is zero and the magnetic field is non-zero.
