You would have to get permission from the relevant stores - and they have no obligation to give that permission.
It is possible for an owner to train their own dog to be an assistance dog - the dog needs to do some physical tasks though and so being shut in a doggy stroller means they won't fulfill that requirement.
We have no option to have emotional support dogs who do not do a physical task, be given the same rights as assistance dogs who do and I don't think anyone involved in assistance dog related work would seek to change that.
If you feel you'd benefit from your dog being with you and you feel their temperament is suitable, then train your dog to do physical tasks - deep pressure therapy, alerting you that you're heading for a meltdown, picking up dropped items, leading you out of a store if you're clearly not coping - these are all physical tasks.
It does take some time and some skill to train an assistance dog and I would strongly recommend you use a trainer who has experience, and have your dogs skills externally assessed either in person or via video submissions.
It is not true that assistance dogs and their handlers cannot be asked to leave a venue - if the dog is posing a danger to others via its behaviour, if the handler is clearly not in control and the dog not mitigating that in some way, if there is a genuine danger to the dog - as long as some reasonable alternative is provided.
So if you did not train your dog, and then you decided to take them with you and they didn't behave, you could still be asked to leave.
For everyone else reading this thread, there do seem to be some misunderstandings.
An assistance dog does not necessarily need to be provided by a charity, let alone an ADUK reg. charity.
The rights to public access apply to ALL assistance dogs trained to perform physical tasks - not just Guide Dogs, and this includes owner trained dogs.
Assistance dogs do NOT have to wear a jacket/vest declaring themselves as such, though it is recommended they do (this is because unfortunately many AD handlers have suffered discrimination as a result of their dog being identified as an AD).
'In Training' dogs do not have public access rights, however a dog can fulfill the basic access requirements of being safe in public and mitigating some elements of someones disability and therefore be 'qualified' for public access, but still be in training for other elements of their work.
Assistance Dogs are still DOGS and not robots, they will look, occasionally sniff something, they may flinch or startle but should bounce back confidently, they may bark if startled or another dog barks at them but again should bounce back fast. They may also bark, climb on their handlers lap or 'jump up', lead their owner away or out of a building or prevent their owner moving (say crossing a road) as part of their physical tasking.
You may NOT ask what an assistance dog does nor what disability their handler has (in the US the rules are different and you can ask what tasks the dog does).
Someone having an allergy to dogs is not a reason for an assistance dog handler to be denied access - the only exception to this rule is for taxi drivers, who will need to register a medical certificate with their licencing authority.