Sadly, if there are double yellows there, I doubt that they would be willing to allow a parking bay - whether for a disabled or able-bodied person.
Of course, it depends on whether the double yellows are there because it's simply dangerous to park there or just to avoid a big bottleneck on a busy road with cars parked in a continuous mile-long line blocking one side. I know others may disagree, but I have my strong suspicions that councils often deliberately put double yellows down on roads near their own pay car parks - no doubt they claim it's to 'ease the access route' to the car park, but revenues might just come into it somewhere too....
However, I think if you are going to apply, your best bet would be to get together with the other disabled resident and submit a joint or 'mirror' applications at the same time. If the council does rule it safe to put in a disabled space, I'd have thought they wouldn't object to two of them. The request may possibly also carry more weight if there is identified a need for disabled parking for residents as opposed to a space for a disabled resident, if that makes sense.
The last thing you want is for them to instal just one space and both of you becoming constant competitors for it, with the 'loser' feeling aggrieved. You don't want to become 'enemies' through the fault of neither of you, when as neighbours sharing similar needs, you should be united.