Submitting form in wordpress plugin

Reviewing wp_dropdown_users() there is apparently a 'selected' parameter. So in theory, you could probably pass the posted value in here.

// set to the posted value, but you might be able to do this differently, depending on how you are storing the data
$selected = isset($_POST['author']) ? $_POST['author'] : false;

$args = array( 'name' => 'author',
               'selected' => $selected );

wp_dropdown_user( $args );