There are various ways dark matter could acquire mass that have nothing to do with the standard model weak force. For example, there are theories involving a hidden sector- particles that do not interact with the standard model gauge bosons at all, but have their own interactions.
Note that the Higgs mechanism is not required for all mass generation in the standard model. The massive gauge bosons acquire their mass through the Higgs mechanism, but there are models where the fermionic masses are acquired through different mechanisms. The source of mass for neutrinos in particular is unknown.
Without knowing what dark matter is, it is of course impossible to determine how it acquires mass.