How to get user ID’s from multiple usernames?

Your call to get_user_by() has a small hiccup in it. It needs to be login rather than user_login

// Search these Usernames
$usernames = array('user1', 'user2');

// Fetch the User IDs
$prof_ids = array(); 
foreach ($usernames as $prof_id) {
    $user = get_user_by('login', $prof_id); 
    $prof_ids[] = $user->ID;
}

// WP_User_Query arguments
$args = array(
     'include' => $prof_ids,
);