Brockswood Animal Sanctuary, in Sedgley, launched an appeal after the animals were stolen overnight into Tuesday.
Staff believe the centre, in Catholic Lane, was targeted as it remains closed amid the continued coronavirus lockdown. coro
They are pleading for anyone with information to get in touch to find them before they are killed.
Workers came onto the site on Tuesday morning to find that two turkeys called Trevor and Clarissa and three sheep called Rose, Florence and Minnie were gone from their enclosures.
It is thought the intruders must have climbed a six-foot high fence to get inside the sanctuary before heaving the turkeys and sheep back over the top.
Sire manager Neil Swann says other animal centres and sanctuaries have been targeted for similar thefts since the lockdown started.
Mr Swann said: "Overnight five animals had been stolen. The people who took them entered the site by climbing over a six foot high fence.
"They lifted each of the animals out of their enclosures over four foot fences, before taking them back over the fence they came in through.
"We know this as there was sheep fleece stuck to the top of the six foot tall fence, lifting these animals is not an easy task.
"Although we do have CCTV we are a 16-acre site, so it doesn't cover the whole of it and the people who came in took a convoluted route to avoid the CCTV we have.
"We think that the animals are being used for food and we are not the first place to have had animals like this taken since the start of lockdown, it seems there is a spate of it in the West Midlands."
The incident has been reported to West Midlands Police who are beginning investigations.
Trevor is a slate grey turkey and Clarissa is a Norfolk black.
Rose, Florence and Minnie are all Soay type ewes; Rose and Florence both have curved horns and Mini has short stubby horns. All are quite skittish.
Brockswood posted the appeal to its Facebook page, which has already been shared by more than 1,500 people.
Supporters of the sanctuary have asked if a JustGiving page can be set up so that they can donate money for the site to improve its security system to prevent further thefts.
Neil added: "We are really pushing it online as are our supporters, we really hope they are found before they are used for food.
"Covid-19 has hit us hard and knocked a hole in our fundraising, we would like to say a huge thank you to all of our supporters who have kept us going during this."
Anyone with information that may help trace the animals should contact Brockswood on 01902 884390, West Midlands Police, Dudley Council or Environmental Services.