Submit is an event listener. You will need to define a function that sends collects and sends the data to your php file. POST might be a good option. Something like this might help get you started.
$("#sendMessageButton").click(function(){
var email = $("#email").val();
$.post("contact_me.php", email, function(response) {
console.log(response)
})
.fail(function(error){
console.error("error:", error)
});
});
Is there are specific reason for using php? If it is not a requirement for your project, shiny's built in validation (https://shiny.rstudio.com/articles/validation.html) might be a good alternative. If the form has sensitive information, it might be a better option to process the data within the shiny server.
Hope that helps!