What is the best way to update the order summary page as soon as checkout page load?, I want to call LiveOrder which should update the order summary page, is there any extensibility API available?

Go for scriptable cart if you will update something for shipping cost or the order itself. If this is for displaying purposes only and custom fields related, then modify the templates associated to it. What exactly you want to achieve?

No I want to update the tax amount which is already I am doing using scriptable cart, I need something with that when check out page get load, order summary should get updated

The summary has a listener for changes on the amount, shipping cost and taxes, if you need to update that, you need to save the LiveOrder.Model. you can try that on the console or click a different shipping method to trigger that and you will see that your changes get reflected. So you need to use in your code a .save but that would be really tricky, otherwise you will see that for sure in the next step.