Yes, you can use the keyup event, and even better would be the input event, since it catches pasting and other types of inputs.
Inside that event handler, you can just check the inputs value, and make a condition that returns a boolean, that can then be used to set the disabled property of a button
$( "#target" ).on('input', function() {
var val = this.value;
$('#button').prop('disabled', val === "some value")
});
And it will update as the user types
In your case, it looks like you want to check three inputs, and if they all have a value, you enable the button, and you'd do that like this
var elems = $('#InputFName, #InputLName, #InputAge').on('input', function() {
var bool = elems.filter(function() {
return this.value.trim() !== "";
}).length !== elems.length;
$('#BtnNext').prop('disabled', bool);
});
$(document).ready(function() {
// Hide the inputs
$("#InputFName").hide();
$("#InputLName").hide();
$("#InputAge").hide();
$("#BtnNext").prop("disabled", true);
// When the user clicks the First name button
$("#BtnFName").click(function() {
$("#InputFName").show();
$("#InputLName").hide();
$("#InputAge").hide();
});
// When the user clicks the Last name button
$("#BtnLName").click(function() {
$("#InputFName").hide();
$("#InputLName").show();
$("#InputAge").hide();
});
// When the user clicks the Age button
$("#BtnAge").click(function() {
$("#InputFName").hide();
$("#InputLName").hide();
$("#InputAge").show();
});
var elems = $('#InputFName, #InputLName, #InputAge').on('input', function() {
var bool = elems.filter(function() {
return this.value.trim() !== "";
}).length !== elems.length;
$('#BtnNext').prop('disabled', bool);
})
});
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div class="Container">
<h3>User Form</h3>
<p>Fill out details:</p>
<button id="BtnFName" class="SrcButton">First name</button>
<button id="BtnLName" class="SrcButton">Last name</button>
<button id="BtnAge" class="SrcButton">Age</button>
<br />
<!-- Input Container -->
<input id="InputFName" placeholder="First Name" />
<input id="InputLName" placeholder="Second Name" />
<input id="InputAge" placeholder="Age" />
<!-- Next Button Container -->
<br />
<button id="BtnNext" class="BtnNext">Next</button>
</div>