The answer to this depends to a great degree on whether you are thinking in terms of "speech" as meaning a specialized form of public discourse within the theoretical constraints of postcolonial theory or whether you are talking about "speech" as more commonly understood.
In the ordinary sense, of course subalterns can speak. Members of all classes of society can talk with their friends and family, chat with staff at shops, or otherwise engage in ordinary forms of daily communication.
There are, however, constraints on the speech of the subaltern. In Nigeria, the language of power was that of the British imperial colonizers. Thus the Igbo had only the choices of speaking an alien language or speaking one which could not communicate with the new centers of power.
The next constraint was access to powerful speaking positions. The pulpits of the Christian churches, the news media, and the government were increasingly controlled by the British, and thus the subaltern had limited access to those platforms which could have disseminated ideas broadly. The speaking within the Igbo clan structures, which was an instrument of power before the arrival of the British, becomes increasingly marginalized; the type of power Okonkwo seeks within the traditional clan structure itself is becoming subordinated to the new colonial powers.