Form Data not getting stored in WordPress database

According to WordPress Codex about the template_redirect hook:

This action hook executes just before WordPress determines which
template page to load.

So, you are redirecting the user to another page before the template is even loaded, which your form won’t load at all. You can use another hook which runs after template_redirect, such as wp_footer.

You can also redirect the user right after inserting the form in the database, by changing your code to:

<?php
    if(isset($_POST['submitnow'])) {
        global $wpdb;
        $wpdb->insert( 
            'wp9c_treetweet', 
            array( 
                'name' => $_POST['aname'], 
                'amt' => $_POST['aamt'],
                'dept' => $_POST['adept']
            ), 
            array( 
                '%s', 
                '%d',
                        '%s' 
            )

        );
        wp_safe_redirect(home_url( '/checkout/' ));
        exit();
    }
?>

This way, if the form is inserted, the user will be redirected afterward.