The Unite union said members had backed action following a "systematic campaign to undermine workers" by employer Babcock Marine.
It will now consult members over the timing and form of any action.
Babcock Marine said it was disappointed with the result of the ballot.
Unite said its members voted 96% in favour of strike action and 99% in favour of industrial action short of a strike.
The union represents a wide range of workers at the two bases, including cleaners, housekeepers, logistics operatives, drivers and mail staff.
They also have members in radiation monitoring and strategic weapons support roles.
The Faslane base on the Clyde is home to the UK's fleet of Trident nuclear submarines.
Unite leaders claimed the company had withdrawn facilities for shop steward meetings and was "failing to meaningfully consult and negotiate with the trade union".
Regional officer Stephen Deans said: "Babcock Marine has engineered a complete breakdown of normal relations between workers and management at Faslane and Coulport.
"Our members believe the company is deliberately trying to undermine their rights, so jobs and service quality can be cut through more outsourcing. It's all about money."
In a statement, Babcock said:  "We regularly engage with our trade unions on a number of topics and view any concerns that are brought to our attention as a matter of importance.
"We are disappointed with this ballot result, as earlier this week we met with Unite and Acas where we had constructive discussions and are due to meet again at the end of this month."
In 2014, Unite members staged a mass walkout at the Faslane and Coulport bases in a dispute over pay.
Civilian staff at the Faslane and Coulport naval bases have voted in favour of industrial action in a dispute over workers' rights.