Webber, 31, joins the club as a free agent after leaving Doncaster this summer and has been training with the Reds throughout pre-season.
The former Sheffield United and Leeds striker will be reunited with Stanley boss James Beattie, whom he once partnered in the Blades' attack.
"First and foremost since I've been here I've enjoyed it," said Webber. "That's a major factor in me being here. There are a good set of lads here and I'm excited to play for the boss.
"I think that good things are happening here, the DNA of the club seems to be on the up and I'm excited to be part of it.
"I played with James when we were strike partners at Sheffield United so there's a lot of history and he is a friend of mine. That's where the initial contact came from.
"I've been a little while out of football for one reason or another and it's ideal for me to come back in, get my form back and enjoy my football again."