It depends on what sort of resources you can afford. Test Automation, done well, is not a part time effort. If you can't afford to have a Toolsmith on your team to work on the Automation Framework you will need help, or expect it to take a long time, or expect it to take effort to keep it going as it gets reprioritized. While most Testers do not spend a lot of time coding and learning to code, many do and they can help provide the assistance, and boost, your team may need. I'm no Developer but I do know HOW to make a Test Framework and get it working, with input from Developers as needed. After all many Developers did not know how to code at some point, they needed to learn how to do it as well. It's also a skill that many Testers need to work on, as the thought process is different but you can usually make a very effective Framework utilizing Coding Skills and Practices with Testing Skills and Practices.
If you want a Framework that does what YOU want, QA needs to own it. QA needs to run it, and QA needs to direct it. You should get buy in from your Developers as well, some Code Review (this is code after all) will be good now and again not just on the Test but on the Framework as well. Make it robust, and most of all make it useful. Other teams should be able to run it, even Developers should have a way to do some localized Smoke Testing in their environments to get their buy-in, and show the usefulness of the Framework for them - get them catching issues early on. The last Framework I used (a BDD Web Driver that used SpecFlow) was able to be used by QA, DEV and the BA's - I just had to show them how to run it. I am not a Developer but I do a lot of automation.
It's good you have the Developers working on Unit Tests, and in the Framework, but remember their goal is to Develop a solution to a problem and your role is to show that their solution solves the problem.
While the browser add-ons are useful, look at something like WebDriver for Selenium and another tool like Cucumber or SpecFlow. Build a framework that will allow you to run a suite of tests, or a select few, so you can cover Regression, Functional and other tests as needed. You also want reports coming from your tests, you should be able to see the historical Pass/Fails for Tests so you know if they are valid, failing only at specific times or show that your pass rate is getting better - management will like that! Any tool takes time to learn, you just need to dedicate that time to your team and give them the resources to improve.