Wire doesn't support exporting chat history (wireapp/wire#116). It is, however, possible to access the decrypted messages in the local storage of a browser. This guide will help you produce a text export once you are logged in https://app.wire.com. All from the browser console.
Open a connection to your browser local storage:
var db
var request = indexedDB.open("wire@production@4278eafb-b21c-4d7a-aad4-2be4fa3a9fa0@permanent")
request.onerror = function(event) {
console.log("Why didn't you allow my web app to use IndexedDB?!")
}