Does it make sense to talk of the inertial mass of a scalar field? By the equivalence principle, it must be equal to its gravitational mass. We know that the scalar field contributes towards the stress-energy tensor, so, shouldn't it have an inertial mass too?
Yes, of course, if you produce a localized concentration of energy carried by a scalar field, it exhibits all the properties that this total energy $E=mc^2$ should exhibit.
It will enter the right hand side of Einstein's equations so it will curve the surrounding spacetime and create a gravitational field. It will be able to convert to other forms of energy so that the total energy conservation law, including the energy of the scalar field, holds.
And finally, it will also act as inertia. For example, a packet of energy carried by a field behaves as a particle. For example, a magnetic monopole may be imagined as a nontrivial localized solution to some field equations including scalar and gauge fields. A magnetic force $F=q_m B$ may act upon the magnetic monopole and the acceleration will indeed be given by $F=ma$ where $m=E/c^2$ is the inertial mass of the magnetic monopole, stored in the energy density of the fields.
This is exactly why it is assumed in the Standard Model that mass is the result of an interaction with a scalar "world potential" (The Higgs field) while for instance the changes in energy/momentum of a charge are due to the electromagnetic vector potential $A^\mu$