Can a magnetic field exist without an electric field present?

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.

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.

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.