No, it's not normal. And, chances are, if the indoor blower quit working while the outdoor section was running, the outdoor section is no longer running (high pressure trip). Your only remedy at this point is to get a heat pump tech to look at it.
Originally posted by idreos
Recently I've heard one of the outside heatpump units operating without its corresponding inside blower. Is this normal?
Building Physics Rule #1: Hot flows to cold.
Building Physics Rule #2: Higher air pressure moves toward lower air pressure
Building Physics Rule #3: Higher moisture concentration moves toward lower moisture concentration.