Shropshire and Telford & Wrekin Councils are taking in families fleeing the Taliban as part of the Government's Afghan Citizens' Resettlement Scheme, which aims to resettle thousands of "at risk" Afghans.
Shropshire Supports Refugees has been collecting donations to support them in the county.
Clothing, bedding, toys and household goods are among the donations which have been offered, but bosses say they have now run out of storage space.
Amanda Jones, director of Shropshire Supports Refugees, said: "There are hundreds of bags of donations.
"We are very grateful. It's been lovely, but we have run out of space and are graciously asking people to pause donations for a while.
"We need to have a chance to process what we've already got.
"We need a large space that's not being used for a couple of days to a week, with easy access, so that we can do this.
"If anyone can help we are asking them to please get in touch.
"If we do still need more donations afterwards we will let people know."
Anyone who may be able to help can email firstname.lastname@example.org
However, donations are still being accepted in Telford and Wrekin, where a total of eight Afghan refugees have already been resettled.
Telford & Wrekin Council says it remains in contact with the Home Office to source accommodation for another 17.
A council spokesperson said: “We’d like to say a big thank you to residents.
"We have been inundated with offers of support, both in terms of welcome messages but also physical support including cash, food, toiletries, warm clothing, baby items and toys.
“We’re working with local organisations to help borough residents in need, alongside those fleeing a crisis and ensuring all donations are coordinated through Telford & Wrekin Interfaith Council and Telford Crisis Support.”
People can contact Telford & Wrekin Interfaith Council by emailing email@example.com or phoning 07545 023519 regarding donations of clothing or blankets.
Donations can be dropped off at the Telford & Wrekin Interfaith Council Community Hub, 19 New Street, Wellington, with advance notice given.
Alternatively, contact Telford Crisis Support by emailing firstname.lastname@example.org or calling 07878 296958.
Donations of food and toiletries can be made at a number of donation points across the borough.
The final UK flight carrying troops and diplomatic personnel left Kabul airport at the weekend, bringing to an end Britain’s 20-year campaign in Afghanistan.