The Hammers finished 10th in the Premier League last time but so far this term have scored just eight goals in 10 matches as they struggle in the bottom half of the table.
Saturday's goalless draw against Aston Villa highlighted how much they are missing the striking prowess of the injured Andy Carroll.
There is no sign of Carroll returning anytime soon, but Allardyce has challenged the rest of his squad to fill his size 12 boots.
Allardyce said: "The problem is our goals were shared out last year. All the players scored. Kevin (Nolan) got 10, Andy (Carroll) got eight and everybody else got two, three, four, one, whatever it might have been and that's why we finished 10th.
"At the moment none of those players are scoring when they get an opportunity. It's not just the front end. It's all of the players not contributing in goal-scoring terms as much as they did last season.
"If we get that right we'll start getting three points instead of one."
The news on Carroll is not encouraging. There is no date for him to resume full training in his recovery from a troublesome heel problem and when he does it will be a month before he can be an integral part of the Hammers challenge.
Allardyce, who once again employed his so-called 4-6-0 formation, said: "When Andy Carroll gets back to full training he's probably the best part of four weeks before he gets on the field in the first team because he has to start a four-week pre-season, which is the minimum we could possibly do."
Considering Villa have scored nine league goals, just one more than the Hammers, it was no surprise the match finished goalless.
Despite West Ham dominating territory and possession, Villa boss Paul Lambert was encouraged, however, by the way his side carved out the game's clear-cut opportunities. Christian Benteke hit the bar with a header and saw another effort well saved by West Ham goalkeeper Jussi Jaaskelainen, while Andreas Weimann also saw an early effort blocked.
Lambert said: "We had the three best chances of the game, on another day they would go in.
"I can't be too hard on them, we are creating chances. We would be worried if we weren't creating chances but Benteke was always a threat, so was Weimann.
"I've got no worries about us taking chances. We played with a back three and went to win the game. We had two up top and you can't get over-worried about how someone else plays. You respect Sam's way of playing. But our resilience is a lot stronger. We looked a threat."
For Allardyce, the primary focus was halting the dismal run at Upton Park which had seen the Hammers slump to three straight defeats.
"The one thing we did right was make sure we didn't concede and stop the run of defeats we have unusually suffered so early in the season," Allardyce said. "The clean sheet was very important.
"We have to keep working at it and believing. Sooner rather than later the ball will end up going in the back of the net for a three-point victory rather than a draw. That will be a big relief to us.
"We've limited Aston Villa to eight crosses only into our penalty area and we've put 38 balls into the opposition's box. It's all about somebody putting the ball in the back of the net when you get so much opportunity. Unfortunately it's our Achilles heel at the moment."