As I said in comments:
INPUT is a void element (like <br> etc...). Closing it with </input> is invalid markup.
Therefore, use a BUTTON element
<button type="button">Apply for job <i class="fas fa-chevron-right"></i></button>
<button type="button"><i class="fas fa-chevron-left"></i> Go back</button>
Using an icon inside the Value of an INPUT
JavaScript solution
If you really cannot change the markup from <input> to <button> element, than you could define the font-family with the desired fallbacks right in CSS, and concatenate the current INPUT value to the unicode representation of the Font Awesome icon el.value += " \uf054":
Example using both Google fonts and Font Awesome:
const addChevron = el => el.value += " \uf054";
document.querySelectorAll('input.button').forEach(addChevron);
.button {
font-family: 'Montserrat', 'Font Awesome 5 Free', sans-serif;
font-weight: 900; /* Needed for font awesome to work... */
}
<link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/all.min.css" rel="stylesheet"/>
<link href="https://fonts.googleapis.com/css?family=Montserrat&display=swap" rel="stylesheet">
<input class="button" type="button" value="Submit to win!">
If you use some Google font, or you want to inherit the font, no worries, pick one:
font-family: 'Some Google Font', 'Font Awesome 5 Free', sans-serif;
font-family: inherit, 'Font Awesome 5 Free', sans-serif;
font-family: 'Font Awesome 5 Free', sans-serif;
you get the idea.
HTML solution
If you don't want to use JavaScript than you could use the unicode representation right inside the value attribute: 
.button {
font-family: 'Font Awesome 5 Free', sans-serif;
font-weight: 900; /* Needed for font awesome to work... */
}
<link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/all.min.css" rel="stylesheet"/>
<input class="button" type="button" value="Submit to win! ">
<br>etc...). Closing it with</input>is invalid markup.