How to dispatch php event with ajax?

You need to make ajax call to a custom controller action and do the dispatch event there as below:

In app/etc/modules/MyCompany_Ajax.xml

<config>
  <modules>
    <MyCompany_Ajax>
      <active>true</active>
      <codePool>local</codePool>
    </MyCompany_Ajax>
  </modules>
</config>

In app/etc/code/local/MyCompany/Ajax/etc/config.xml

<config>
    <modules>
        <MyCompany_Ajax>
            <version>1.0.0</version>
        </MyCompany_Ajax>
    </modules>
    <global>
        <frontend>
            <routers>
                <mycompanyajax>
                    <use>standard</use>
                    <args>
                        <module>MyCompany_Ajax</module>
                        <frontName>mycompanyajax</frontName>
                    </args>
                </mycompanyajax>
            </routers>
        </frontend>
    </global>
</config>

You can send the ajax request to a controller and dispatch the event from there:

#app/code/local/MyCompany/Ajax/controllers/AjaxController.php
<?php
class MyCompany_Ajax_AjaxController extends Mage_Core_Controller_Front_Action
{
    public function paymentAction()
    {
        $paymentMethodStatus = $this->getRequest()->getParam('is_active', false);
        //The $args variable may differ based on what values you are expecting in your observer.
        $args = array ('is_active', $paymentMethodStatus);
        Mage::dispatchEvent('payment_method_is_active', $args);
    }
}

In you template file:

$("#paymentCheckBox").live("click", function(){
    var isChecked = $(this).is(":checked");
    $.ajax({
        type: "POST",
        url: mycompanyajax/ajax/payment,
        data: { 'is_active' : isChecked},
        success: function(data) {
            //Handle if you return something
        },
        error: function() {
            alert('Something went wrong');
        },
        complete: function() {
            alert('Its done');
        }
    });
});