The Government guidance on exemptions suggests there is no requirement for evidence for exemption, therefore it should be sufficient for an individual to self-declare this.
Practices are not required to provide letters of support for those who fall under the list of exemptions.
Similarly, practices are under no obligation to provide support letters for anyone who does not fall under the list of exemptions but considered them to have reason to be exempt.
https://www.gov.uk/guidance/coronavirus-covid-19-safer-travel-guidance-for-passengers#exemptions-face-coverings