The Communication Workers Union (CWU) balloted about 4000 of its members who work in the Crown post offices – the larger sites usually based in high streets.
There was an 88% majority in favour, in a turnout of 75%.
In Scotland, post offices in East Kilbride and Cumbernauld are among five earmarked to become franchises in shops.
The five branches, which employ 49 people, may be moved as the Post Office seeks partners for the controversial venture.
The other three Scottish branches affected are in Perth, Alloa and Stornoway. Across the UK, partners are being sought for 70 branches.
Dave Ward, CWU deputy general secretary, said: "Our Post Office members have spoken loud and clear in this ballot. They are fed up with being treated like second-class workers by the Post Office and they want their concerns to be listened to.
"There hasn't been so much as the pretence of negotiations from Post Office Ltd, and that must change if strike action is to be avoided.
"We are now asking the company to come to the table to discuss pay and job security for these workers and avoid the need for strike action."
Andy Furey, CWU official for Post Office members, added: "The Crown network brings significant revenues into the business and is crucial to the effective delivery of many products and services.
"The company has given pay rises to other employees and now it's time they played fair and awarded a pay rise to Crown office staff.
"Job security and the future of the network are the elephant in the room here, and we want to see the Post Office acting as a responsible and fair employer, not treating parts of the workforce differently and gambling with services which communities rely on."
Kevin Gilliland, network and sales director at the Post Office, said: "We are very disappointed with the CWU's reaction to the transformation programme for the Crown network. Our offer of cash payments recognises the contribution being made by our people to the turnaround of the business.
"There is absolutely no room for manoeuvre. These payments are fair at a time when our Crown network is operating at a loss. Our door remains wide open to discussions with the CWU as to how we get the first of these payments into pay packets as quickly as possible.
"The Crown network is making a loss of £40 million per year. This is public money. We want to keep Post Office branches on high streets across the UK and the best way to do this is for 70 of our Crown branches to partner with a suitable retailer."
The Post Office said it had made an offer of cash payments totalling up to £3400 to be paid before April 2015.