How can I access jQuery plugin function?

Maybe you can change togglePushy to something like this:

$.togglePushy = function(){ 
    body.toggleClass(pushyActiveClass); //toggle site overlay
    pushy.toggleClass(pushyClass);
    container.toggleClass(containerClass);
    push.toggleClass(pushClass); //css class to add pushy capability
}

Now you should be able to access it as $.togglePushy()

I know not the best of the solution. But you can do this by keeping your functions outside the domaready event handler like following.

function togglePushy(){
    body.toggleClass(pushyActiveClass); //toggle site overlay
    pushy.toggleClass(pushyClass);
    container.toggleClass(containerClass);
    push.toggleClass(pushClass); //css class to add pushy capability
}

function openPushyFallback(){
    body.addClass(pushyActiveClass);
    pushy.animate({left: "0px"}, menuSpeed);
    container.animate({left: menuWidth}, menuSpeed);
    push.animate({left: menuWidth}, menuSpeed); //css class to add pushy capability
}

function closePushyFallback(){
    body.removeClass(pushyActiveClass);
    pushy.animate({left: "-" + menuWidth}, menuSpeed);
    container.animate({left: "0px"}, menuSpeed);
    push.animate({left: "0px"}, menuSpeed); //css class to add pushy capability
}

$(function() {
    var pushy = $('.pushy'), //menu css class
        body = $('body'),
        container = $('#container'), //container css class
        push = $('.push'), //css class to add pushy capability
        siteOverlay = $('.site-overlay'), //site overlay
        pushyClass = "pushy-left pushy-open", //menu position & menu open class
        pushyActiveClass = "pushy-active", //css class to toggle site overlay
        containerClass = "container-push", //container open class
        pushClass = "push-push", //css class to add pushy capability
        menuBtn = $('.menu-btn, .pushy a'), //css classes to toggle the menu
        menuSpeed = 200, //jQuery fallback menu speed
        menuWidth = pushy.width() + "px"; //jQuery fallback menu width



    menuBtn.click(function() {
        togglePushy();
    });
});

Then call the togglePushy() directly.

Your togglePushy() function is defined only in the context (scope) of the jQuery closure (anonymous function). Define it in the global scope instead (elsewhere in the file and not as a "function parameter").

If you need variables from that scope, encapsulate them in a namespace (an object), or declare it as window.togglePushy() = function() { /* ... */}.

or

$(document).ready(function() {
  ...
  .....
  window.togglePushy()= function() {
    //to do
  } 
});

or

$(document).ready(function(){
    $('button').click(function(){
        $.togglePushy();
    });
});