Can a pet be trained to be a support dog?

It depends on the type of dog you have, but they all can be trained. You can start with some simple exercises at home and get professional help from a company or assistant. I got some good inspiration at cutepetcare.com/.../. They have a lot of articles on the matter, and I'm sure you can find something helpful there. At the same time, seeing your kind little dog made me so happy. Animals are real angels; if you love them, you'll get ten times more love from them.

You can train your pet dog as an assistance dog. I'm going to train my Labrador, but we should complete basic dog courses before to be able to begin the assistant dog course.

You might want to think carefully about whether this is going to give you what you want and need. Equality and Disability law supports the right of a person related to their disability, so if bought to court, it may be incumbent on the dog’s owner to demonstrate how their dog legitimately supports their mobility and or independence. This would apply whether the dog is trained by a member of ADUK ( Assistance Dog UK) or not.
There are many individual assistance dog trainers and non-accredited organisations offering a service, but it is the responsibility of the individual to do their own research into the trainer’s/organisations qualifications, training techniques and understanding of the appropriate disabilities and conditions.

Some useful  links
http://www.sherlockhounds.org.uk/assistance-dog-law/

You could get some advices on  Facebook

Assistance Dog for Young People UK (charity and owner trained) https://www.facebook.com/groups/1778990935498855/?multi_permalinks=2846160655448539%2C2851041601627111&notif_id=1581703018357430&notif_t=group_activity

or

Assistance and Service Dogs UK (Owner Trained) https://www.facebook.com/groups/1719619064740514/