That is an excellent question, and not a trivial one to answer.
To try to provide an answer, it is good to go back to a definition of inertia, especially as such a term is not used that much anymore. We can see inertia as the tendency for a given object to resist to a change in motion. Then, we can move one step further and definite an inertial frame of reference as a system in which the inertia of the object is equivalent to its mass. Inertia is thus somewhat a function of the object mass.
Things get more complicated when we consider speeds close to the speed of light and relativity: inertia now gets affected by the object mass and velocity (speed and direction). In other words, inertia is not anymore something intrinsic to the object. We can show (by making use of special relativity) that the object's inertia is in fact larger than or equal to the mass (we don't have a strict equality anymore).
From this last point, a massless particle can thus have inertia.
I am not 100% sure this answers fully your question, but at least it should provide seeds to discuss it further, if needed.
Cheers!