Brackenhill allotments, just off Wakefield Road, were targeted between Friday, January 16 and Sunday, January 18.
Scrap metal and gardening tools were stolen during the raids.
Insp Geoff Carter, of South East neighbourhood policing team (NPT), said: “These raids were obviously targeted as we had seven reported over the same weekend.
“The suspects only stole scrap metal from one plot but they caused considerable damage to seven sheds on the site.”
Vandals also trashed fencing and stole clothing from one of the allotments.
Insp Carter added: “I would appeal for anyone with any information or who saw anyone acting suspiciously around the allotment site over the weekend to get in touch.
“If your allotment has been damaged and you haven’t reported it to us then I would urge you to get in touch.”
The allotment site, run by Ackworth Parish Council, has 100 plots.
Sgt Sarah Degnan, also of South East NPT, said it was a rare crime. She said: “It’s strange that we had seven plots vandalised over the same weekend.
“We would ask the allotment holders to remain vigilant and if they notice anything suspicious to contact us.”
Anyone with any information should call South East NPT on 101 or call Crimestoppers, anonymously, on 0800 555111.