Do Chrome extensions have access to local storage?

Yes, it is. Going over a full walkthrough of how to do this would probably exceed the length of a reasonable StackOverflow answer, so I'll refer you to this very extensive tutorial by Rajdeep Dua.

The relevant code would look like this:

// Store item in local storage:
function setItem(key, value) {
  try {
    log("Storing [" + key + ":" + value + "]");
    window.localStorage.removeItem(key);      // <-- Local storage!
    window.localStorage.setItem(key, value);  // <-- Local storage!
  } catch(e) {
    log("Error inside setItem");
    log(e);
  }
  log("Return from setItem" + key + ":" +  value);
}

// Gets item from local storage with specified key.
function getItem(key) {
  var value;
  log('Retrieving key [' + key + ']');
  try {
    value = window.localStorage.getItem(key);  // <-- Local storage!
  }catch(e) {
    log("Error inside getItem() for key:" + key);
  log(e);
  value = "null";
  }
  log("Returning value: " + value);
  return value;
}

// Clears all key/value pairs in local storage.
function clearStrg() {
  log('about to clear local storage');
  window.localStorage.clear(); // <-- Local storage!
  log('cleared');
}

function log(txt) {
  if(logging) {
    console.log(txt);
  }
}

Chrome also supports the HTML5 Web Database spec. This gives you a local SQL database, so you can do more complex things than simply storing name/value pairs in Local Storage.

Even simpler than that:

To read:

    var myStoredValue = localStorage["TheKeyToMyStoredValue"];

To write:

    localStorage["TheKeyToMyStoredValue"] = myNewValueToStore;

To get rid of:

    delete localStorage["TheKeyToMyStoredValue"];

The current chrome version has local storage.

I have used it myself. You can use modernizr to detect whether the browser supports it or not. I have written a solution for a client where I do a fallback to cookie if no local storage exists, but this shouldn't be a problem for extensions.