There is actually no legal basis for either supermarkets or Police to be entitled to see proof of medical exemption. Businesses can actually be at risk of civil action for disability discrimination if they refuse entry without proof.
This article explains it well
The NPCC (The National Police Chiefs Council) has responded by saying police have “clear guidance” on the exemptions to face coverings rules.
“Officers will have conversations with people to understand if they are exempt,” said an NPCC spokeswoman.
"Whilst officers will be inquisitive to establish the facts, we are clear that there is no requirement to provide medical proof of an exemption."
The government website states:
'Exemption cards
If you have an age, health or disability reason for not wearing a face covering:
you do not routinely need to show any written evidence of this
you do not need show an exemption card
This means that you do not need to seek advice or request a letter from a medical professional about your reason for not wearing a face covering.
However, some people may feel more comfortable showing something that says they do not have to wear a face covering. This could be in the form of an exemption card, badge or even a home-made sign.
Carrying an exemption card or badge is a personal choice and is not required by law.'