Reload Chrome extension tabs

Is there some way to get the tab id's of only the tabs that are part of my extension?

This really depends on what you mean by "part of my extension".

If you mean tabs that are displaying a page that is contained within your extension you can do the following;

chrome.tabs.query({}, function (tabs) {
  var myTabs = [];
  for (var i = 0; i < tabs.length; i++) {
    if (tabs[i].url.indexOf(chrome.extension.getURL('')) === 0) {
      myTabs.push(tabs[i].id);
    }
  }
  console.log(myTabs);
});

If you want to access the DOM of your tabs instead, it gets even easier;

var myTabs = chrome.extension.getViews({type: 'tab'});

With access to the DOM you can simply iterate of each view (DOMWindow) and refresh each page;

for (var i = 0; i < myTabs.length; i++) {
  myTabs[i].location.reload()
}