Sir Tom died aged 100 last week after being admitted to Bedford Hospital having tested positive for Covid-19.
His family said due to medication he was receiving for pneumonia, he was unable to be vaccinated.
Sir Tom raised more than £32m for NHS Charities Together by walking laps of his Bedforshire garden during the first lockdown.
He was subsequently knighted by the Queen in July in a special ceremony at Windsor Castle.
Following the complaint made on Friday (February) 5 the suspect was arrested and charged in connection with communication offences.