Anyone with a blue badge is exempt from the congestion charge wether you have a Motobility car or privately own one.
Until 5 years ago, I lived in London and had a blue badge but owned my own car. All you have to do is register with TFL and provide proof of your badge. You are given an account and are allowed to have 2 cars actively receiving 100% discount at the same time. So I had my car and one of my daughters cars as active and could switch cars daily or ongoing.
Even though I moved away from England 5 years ago, I still have my account active, having renewed it with new badges and have my daughters cars on so they don’t have to pay the charge.