All materials do not act as conductors because of absence of free electrons.
To answer this question well, lets understand what an electric conductor is.
A conductor is any substance that conducts electricity. And this can be done by displacing charged particles.
In most of the cases these charged particles are electrons. Metallic substances have free electrons and hence they act as conductors.
In ionic conduction, the 'charged particles' mentioned above are ions in a solution.
But since all the substances do not have free electrons or do not possess free ions, all substances are do not act as conductors.
Some examples of non conductors of electricity are:
Wood, paper(most of the organic compounds), Pure water, air, and so on.