Question

electrons must be able to flow through a material in order for the material to conduct electricity

Answer 1

Yes — for a material to conduct an electric current there must be mobile charge carriers that can move through it when a potential difference is applied.

Key points:
- In metals the mobile carriers are electrons in partially filled conduction bands; those electrons can drift under an electric field and carry current.
- In semiconductors both electrons (in the conduction band) and holes (absence of electrons in the valence band) can move and contribute to conduction; their availability depends on doping and temperature.
- In electrolytes and ionic solids the carriers are ions (positive and negative), not electrons.
- In insulators the electrons are tightly bound and there is a large energy gap to the conduction states, so very few carriers are available and conduction is negligible (unless you apply a very large field and cause breakdown).
- Conduction requires not just mobile carriers but also a closed circuit and an electric field or potential difference to drive carrier motion.

A simple relation for conductivity is sigma = n q mu, where n is carrier density, q the carrier charge, and mu the mobility; the current density J = sigma E. Special cases: superconductors conduct with zero resistance because carriers form a coherent state with no scattering.

So the short answer: yes — electrons (or other charge carriers) must be able to move through the material for it to conduct electricity.