Transport and General Workers Union (TGWU) members based at Bristol, Northwich and Wakefield voted by 918 to 289 to take the action in a row over job losses. GMB members vote on industrial action today.
The TGWU and GMB unions will meet after the ballot to decide their next move.
Wednesday's announcement follows the planned closure of three distribution depots, at Kent, Bristol and Cheshire, with the potential loss of up to 2,500 jobs. The union says staff were not consulted.
Morrisons claims to have contingency plans in place should it face a strike and said: 'We have 12 other distribution sites operated by third parties and/or represented by other trade unions not involved in these ballots.'
However, the TGWU warned that action raised the prospect of 'serious shortages' on the shelves. TGWU national officer Brian Revell accused the company of refusing to consult with unions and said Morrisons had 'blocked' trade union activity.
Morrisons said it has spoken with Revell and is happy to discuss the ballot result. 'We will offer similar discussions to GMB national officer Jude Brimble once the outcome of their ballot is known,' the company said
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