Most of the info you need is here.
www.gov.uk/government/uploads/system/uploads/attachment_data/file/197719/can-i-get-a-blue-badge.pdf
First thing to get is a letter from the GP supporting the Blue Badge application.
I found our local Council Customer services centre were very helpful, they'll fill the form in for you and even take a photo to use on the back of the badge for ID.
If your relative can get there easily, it may be an option.
Take the doctor's letter and and copies of all the other supporting documents you need which are :-
Their national insurance number
Their driving licence.
Proof of their identification e.g. a birth or marriage certificate, a driving licence, a passport or an ID card
Proof of their address e.g. a letter from their bank or a utility bill
A photograph of them (Our Customer Services lady took it on a camera connected to their computer, so don't worry if you don't have one) They were happy with copies of the documents but I took the originals just in case.
Explain their condition in as much detail as possible Describe how their condition affects their daily life and your mobility. Try to estimate how far they can travel without assistance, (see below, leaning on a person or a trolley for support is not 'without assistance') how long it takes them to travel this distance and how travelling this distance without assistance makes them feel e.g. if it causes pain, makes them feel out of breath or dizzy, etc
Please be careful filling in the above details, older people particularly want to paint the best picture of themselves and if they say they can "walk round the supermarket without help" they'll likely not be given a badge but if the truth is more like this, then describe exactly what happens. e.g. They need a hand getting out of the car and need a wider space to open the car door fully so you can help after you've got the trolley for them to lean on then they can be supported by the trolley whilst they look at a few things whilst you run round the aisles getting all the things that are needed that trip, then take the relative to the courtesy seating whilst you check out, then they lean on the trolley to get back to the car.
Also describe how they manage indoors, do they use a stick or rollator or zimmer frame, need a grab handle on steps etc.
Sometimes, when the form has been submitted, the person it's for will be asked to go for a Mobility Assessment Interview by the Council's trained physio or OT or other trained professional.
It's not torturous and the assessors are very kind. They ask a few questions and in our case observe the person walking back to the car , noting how long it takes, how unsteady they are, if they get breathless etc.
They cannot give you an answer there and then, it's not solely their decision.
We heard a week later we'd been successful, the timeframe will be dependent on how many cases are being dealt with at any one time.
Good Luck
I hope you are all successful. As I've said above, it's how you answer the questions that can make a big difference.