The Hammers have scored just eight goals in 10 league matches this season and, while they ended a dismal run of three straight defeats at Upton Park, the goal drought is becoming more concerning by the week.
Carroll has a troublesome heel problem and there is no date yet for when he might resume full training.
Allardyce said: "When Andy gets back training it's probably the best part of four weeks before he gets back onto the field in the first team. He has to start a four-week pre-season. We'd try to speed that up if we possibly could."
Allardyce believes the lack of goals are not down to the strikers alone.
He said: "The problem is our goals were shared out last year. All the players in our team scored. Kevin (Nolan) got 10, Andy (Carroll) got eight, everybody else got three, two or one and that's why we finished 10th.
"None of those players are scoring goals as they did last year. It's not just the front end, it's all the players not scoring.
"When we get that right, which I think we will, we'll start getting three points instead of one."
Allardyce was pleased with the clean sheet against a side that have also failed to score in their past four matches.
It might have been different if referee Howard Webb had ruled in Villa's favour in a controversial incident in the first half.
Villa striker Andreas Weimann broke clear and appeared to be pulled down just outside the penalty area by West Ham midfielder Ravel Morrison.
Morrison was clearly the last man but instead of giving a foul and producing the inevitable red card Webb waved play on.
Weimann slumped to his knees in disbelief while Villa boss Paul Lambert remonstrated on the touchline.
Lambert said: "I have spoken to Howard and I respect his view and know where he is coming from. If the ball is on the safe side (goal side) it is difficult to win it.
"I respect his decision and the way he saw it. I just thought from my angle it might have been (a foul). I have to respect his answer. Sometimes you get them, sometimes you don't."
Lambert, however, was pleased with his side's fighting spirit which saw Christian Benteke hit the post with a header and go close with a shot just deflected wide.
"It was a big performance," said Lambert.
"We were missing a couple of big players. I don't think West Ham had many chances to score a goal. The lads at the back put in their heads when it mattered and blocked when it mattered and along with goalkeeper Brad Guzan I thought they were all excellent."