The move would create a 20,000 all-seater stadium that will also be used for Northern Ireland international games.
The members voted by a large majority to support a draft agreement between the club and Irish FA.
The running of the stadium would be taken over by a management company.
As owners of the site, Linfield would receive £200,000 in rent annually from the IFA and the club will not have to pay for any maintenance costs.
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"The Irish FA welcomes and is pleased with the decision taken by members of Linfield FC," the governing body said in a statement.
"This is another important step on making the redevelopment of Windsor Park a reality.
"The Association will now meet with our partners - DCAL, Sport NI and Linfield FC to discuss the next step in this process."
In the past, rival Irish League clubs have expressed concern about the money Linfield were receiving for the use of Windsor Park for international matches.
Their current contract with the IFA nets them 15% of revenue from internationals fixtures.
It is expected work on revamping the out-of-date Windsor would take around 18 months. The ground has a current capacity of just 14,500.
"This is a good day for Linfield FC and for football in Northern Ireland," said Linfield president Peter Lunn.
"Our members have shown their support, in a mature and responsible way, for the broad agreement we have reached with the IFA."
(NS/BMcC)
