Suez, which runs waste and recycling collections in the borough, is down by a fifth of the number of drivers they need, say Calderdale councillors.
They require 40 drivers but currently only have 32.
The shortage - part of a national lack of HGV drivers - means some residents' collection are being missed.
Helen Rivron (Lab, Ovenden) said she has had complaints from residents about the situation, particularly when crews have repeatedly missed two weeks of recycling collections.
She said steps are being brought in to address the situation.
"They are doing what they can," she said. "They're trying to get more drivers and to use the staff they have in the most effective way.
"They are trying to make sure no one misses two collections in a row.
"We can't magically produce more HGV drivers and that is a national problem.
"It's important that we keep talking to people and keep them up to date."
She said measures have been put in place with Suez to ensure that where a round is dropped, it becomes a priority for the following week.
She also said it is difficult to recruit staff or use agency staff because other organisations can offer better pay and conditions.
Councillor Mike Barnes (Lab, Skircoat) said recycling collections have been hardest hit.
"The impact is that we are not able to run a full service – around 85 per cent of households are still receiving a weekly collection, although this is no consolation for those where their collection was not picked up."
He added his praise for the collection workers.
"I think they do a really good job in trying circumstances.
"I can't think of a resident who has contacted me who had an issue with the teams. Residents appreciate the job they are doing."
Calderdale Council’s Cabinet Member for Public Services and Communities, Councillor Jenny Lynn, said: “We’re aware of ongoing issues with waste and recycling collections in the borough and apologise for any problems this is causing for residents.
"Service disruption plans are in place to minimise problems as much as possible.
“Unfortunately, the nationwide shortage of HGV drivers is continuing to cause challenges for organisations up and down the country – and waste and recycling collections are not immune from this.
"The shortage is having an impact on collection schedules and logistics across the entire waste and recycling sector, however we are working hard to ensure that residents’ bins are collected as regularly and consistently as possible throughout these challenging times.
“We’re continuing to assess the level of disruption and are in regular contact with Suez to try and look for ways to resolve the issue. "In the meantime, we’re asking residents that if their recycling is not collected on the scheduled day, they should take their containers back in and place back out the following week on their scheduled day. " Alternatively, waste can be taken to local Household Waste Recycling Centres.
“Disruption is also being caused by high volumes of waste being presented for collection and access issues on some routes due to parked cars.
"Please support the crews by ensuring that streets are accessible to service vehicles, this includes avoiding parking around junctions or on both sides of narrow streets. If parking on the road, please also fold in wing mirrors.
“We do understand the frustration that this disruption is causing and thank residents for their patience at this challenging time.
"We know where work is outstanding, so it’s not necessary for missed collections to be reported.
"For the latest service updates, visit https://www.calderdale.gov.uk/v2/coronavirus/service-updates."