Craft Commerce — Update order status from frontend?
This feels a bit dirty, but you can actually do this directly in a Twig template like this:
{% set order = craft.orders.id(123).one() %}
{% do order.setAttributes({orderStatusId: 4}, false) %}
{% do craft.app.getElements().saveElement(order) %}

-- naturally you'd need a way to fetch which order you want to update, but you could easily grab that from your POST data or from the URL or localStorage or whatever.