You can make an ajax call when you detect a click on a radio button. It'll hit a PHP file and you can then insert the vote from there. Check out jQuery Ajax for examples.
Example:
$.ajax({
type : 'POST',
url : '/ajax/vote.php',
data : {
value : radio_button_val
},
beforeSend : function() {
// loading image
},
success : function(data) {
// successful
},
error : function() {
// failed
}
});