Almost 400 Best Food Logistics workers are set to walk out after rejecting a 6% pay offer, which the GMB Union said amounted to a real-terms pay cut.
Some 76% of those who voted were in favour of industrial action.
Best Food Logistics drivers, who also deliver to the likes of KFC, Burger King, Pizza Hut and Zizzi, will announced strike dates imminently.
Nadine Houghton, GMB National Officer, said: "The parent companies of Best Food - Booker and Tesco – are making serious money.
"Shareholders are trousering large dividends, while the people who do the graft are struggling to make ends meet.
"All these workers want is a pay deal that protects them from this crushing cost of living crisis.
"Now, some of the best know restaurants on the UK's high streets will face shortages over Christmas."
A Best Food Logistics spokesperson said: "Best Food Logistics provides colleagues with a competitive and comprehensive reward package, and we are committed to a constructive dialogue with our colleagues and GMB to reach a way forward.
"We will be working closely with our customers to ensure they can continue to get the products they need and remain actively engaged with GMB to reach a positive outcome for our colleagues and customers."