A senior officer at Wakefield Council said that authorities were struggling to get answers from government departments about how they should be planning for life after March 29, the day the UK is currently set to leave the EU.
In West Yorkshire, there is particular concern about possible congestion along the M62 if imported goods are diverted to the port of Hull, which could happen if the UK crashes out without a deal.
Last week, the council's chief executive said the impact of leaving the EU was unknown while the nature of any Brexit deal remains uncertain.
And on Monday, the council's corporate director for business change, Gillian Connolly said the authority was planning "without knowing what we are planning for".
She added that there was "anxiety" among senior regional leaders about the government's responses.
Speaking at an audit committee meeting when the issue was raised, Labour councillor Richard Forster said: "I'm aware of an information document that's been released to local authorities and that's now live, but it's very limited.
"I know there's some (information) in relation to transport, with Hull possibly being used as a port and the knock-on effect on the M62. There's also issues regarding the possible impact on people in our care homes.
"But the government are actually limiting the amount of information they are releasing to local authorities. Is that impacting on what we're actually able to plan for, if we don't have all the information the government holds?"
In response, Ms Connolly said: "I think the anxiety in West Yorkshire is more that there are now lots of meetings with government and lots of questions, but getting answers can be quite difficult.
"I think in Wakefield we're taking a pragmatic approach that we'll do what we can with what we've got."
Ms Connolly said that heavy congestion along the M62 could stop people getting into the city centre for work but that this would represent a "worst case scenario".
She said that "business continuity plans", which the council has written up to help companies adapt to post-Brexit changes would be put into action at that point.
The Department for Communities and Local Government has been contacted for comment.