Amond got off the mark for the campaign after nine goals last season by blasting his spot kick into the top right corner after 43 minutes.
The penalty - Stanley's first to be awarded since April 2011 - was given when Amond's strike partner Karl Sheppard was brought down by Vale goalkeeper Chris Neal, who was shown a yellow card.
The game had been even up until then, with Neal's opposite number Ian Dunbavin pulling off a superb save from a Louis Dodds snap shot in the sixth minute. Dodds fired just over the bar following a deflection midway through the half.
Stanley started to get on top with Luke Joyce coming close and they got their reward shortly before the interval.
It was back to the walls for Stanley after the break as Vale piled on the pressure. Sam Morsy almost levelled in the 56th minute, but a combination of Dunbavin and Joyce scrambled his close range effort away, while the Reds keeper punched Clayton McDonald's fierce strike to safety before Gray had the final word.