Dark matter does have gravitational mass as we know from its discovery. Does it have inertial mass?
The answer depends on the identity of the dark matter.
In the most widely believed scenario, dark matter is composed of "weakly interacting massive particles" ("WIMP"). The adjective "weak" really means that the particles interact via the weak nuclear force. This pretty much guarantees that they interact with the Higgs boson, too: the WIMPs carry the hypercharge or the weak nuclear charge (that's what "WI" in "WIMP" guarantees).
According to current theory, the most likely WIMP is the LSP, the lightest superpartner in a supersymmetric theory. The LSP may be a gravitino in which case its interactions with the Higgs bosons are almost non-existent – gravity is the only relevant interaction; or they may be a "neutralino" (higgsino, photino, or zino – or, in a different basis, higgsino, neutral wino, or bino) and this interacts with the Higgs boson as strongly as the known superpartners of the neutralino, namely the Z-boson or the Higgs boson itself.
So if the dark matter is composed of neutralinos, then it interacts with the Higgs boson. However, the Higgs field isn't the main reason why these particles are massive ("M" in "WIMP"). Instead, the main reason is a supersymmetry breaking mechanism, something independent from the Higgs mechanism which may also be called the electroweak symmetry breaking mechanism. It's due to SUSY breaking, and not the electroweak symmetry breaking, that the LSP is massive. For example, SUSY breaking is needed to make photino (or bino/wino/neutralino that contains a piece of it) massive even though the photon, its known superpartner, is massless.
If the dark matter is made of axions, its interactions with the Higgs field are virtually non-existent. If dark matter is composed of MACHOs or if it doesn't exist at all, it makes no sense to describe its interactions at the level of particle physics because it is a composite object or because it doesn't exist. For example, if dark matter were composed of black holes, they would interact like other black holes – pretty much only gravitationally.
Gravitational mass is the same as inertial mass. If there is dark matter, it has to be cold. Massless particles can only give rise to "warm" radiation. So it has to be massive.