You're confusing the passing of the register to :yank (Ex command) vs. y (normal mode). For the latter, instead of appending the register, it must come before the y command, and with the " prefix: "Ayi" (with register A, yank the inner " double quoted text)
:let @a = '' | g/prop/normal! "Ayi"
Note that as the yank is characterwise, all matches will be ...
Somebody had a similar problem: vim yank all matches of regex group into register.
In your case regex will be prop="\zs.*\ze".
\zs denotes where the match will start in the "substitution" (yanking to register), and \ze where to end.