How can I process a form submission from an HTTP POST request?

you can use admin-post.php to process the form.

you can set the form like that :

<form
    action="<?= htmlspecialchars(admin_url("admin-post.php?action=MY_PLUGIN__my_action"))?>"
    method="POST"
>
    
    <button name="send">go</button>
    
</form>

and then you link the code with that :

add_action("admin_init", function () {
    
    if (    !isset($_GET["action"])
        ||  ("MY_PLUGIN__my_action" !== $_GET["action"])
    ) {
        return;
    }
    
    
    /*  */
    // debug
    var_export($_POST);
    exit();
    /*  */
    
    
    // data processing
    //...
    
    
    // redirection at the end
    wp_redirection(home_url("done"));
    exit();
    
});