The National Housing Federation (NHF), which compiled the figures, wants the Government to exempt vulnerable people from the spare room penalty to be introduced next month.
Opponents say those hit by a benefits cut of up to an average of £637 a year in some parts of Devon and Cornwall will be forced to find a new home – but finding a smaller home is impossible given the chronic shortage of properties.
Catherine Brabner, the NHF's South West manager, said: "The Government's bedroom tax is flawed and will unfairly penalise thousands of people in the South West who have lived in their homes for years, raised families and contributed to their communities.
"The 'one-size-fits-all' approach takes no account of disabled people's adapted homes, of foster parents who need rooms to take children in, or of parents sharing custody who will lose the room for their child at weekends.
"In most areas, there just aren't enough smaller affordable homes for these families to move into to avoid the tax. Many people will find themselves having to move into more expensive privately rented properties – adding to the overall housing benefit bill."
She added: "The high housing benefit bill is because there are not enough affordable homes, so the best way to cut the bill is to build more."
The Western Morning News this week reported that 30,000 people in the wider South West would be hit by the cut to benefits for under-occupancy, according to the official impact assessment.
The NHF, which represents housing associations, estimates 9,170 of those are in Devon and Cornwall, and in turn 5,777 have disabilities. Ministers are determined to bring down Britain's bloated benefits bill, but have given councils more than £150 million to help vulnerable tenants.
Labour has warned the spare-room clampdown will hit a range of social groups, from households with relatives away serving in the armed forces to separated parents who keep a room so their children can stay.
The NHF says the Government's own estimate is that 63% of those affected are disabled, and that funding to offset the impact on the vulnerable will hand them as little as £1.71 a week.
Under the Government's proposals, people in social housing will receive a cut in housing benefit where they are deemed to have spare bedrooms.
Ministers dispute the term "bedroom tax" – arguing it is not a "tax" but a partial withdrawal of a state hand-out.
A Department for Work Pensions spokesman said: "Councils have been given an extra £155 million this year so that they can help their vulnerable tenants, with £30 million specifically targeted towards supporting disabled people who have modified their homes and foster carers.
"We need to ensure a better use of social housing when over a quarter of a million tenants are living in overcrowded homes and 2 million are on housing waiting lists."
