Yes a bad water pump could cause the heater in your vehicle to not work.
If your vehicles water pump fails and is not circulating any coolant through the engine and heater core then the heater will not produce any heat.
But if your water pump is not working then your engine will also overheat because the coolant won't be flowing through the engine properly.
Water pumps rare fail to pump water through the engine because if the water pump is belt driven then there's not much that can fail on the water pump.
The water pump spins on a bearing and as long as it's spinning the coolant should flow through the engine.
If it's an electric water pump then it can fail and cause the engine to overheat and the heater to not work.
Your heater in a car works from the heat of the engine coolant so if not coolant is getting through the heater core then the heater will not work.
But usually a clogged heater core or bad heater switch causes the heater to not work.
If your blower motor is not working then your heat won't work either.