Short answer: No, it does not have any impact at all.
Longer answer:
The ‘Order’ field is a remnant from the early days of WordPress. Its only purpose is to order the pages in the admin for easier content management. Plus, people can use it to display a list of pages in their preferred order, for example in the main navigation. Nowadays custom nav menus are being used everywhere though, so it’s not really useful there. This means nothing in the front end of your website changes when you manipulate the order field.
None of the SEO plugins I know use the order field in any way. Why should they? The order in which URLs are listed in XML sitemaps (which you can submit to search engines) is a) not determined by the order field (none of the SEO plugins I checked use that field anywhere) and b) has no effect on their priority. XML sitemaps have a separate priority field for that, which would be determined by the importance of a page, the frequency of changes to it, and things like that — the menu order has nothing to do with that. And even if it would, Google is not even using the priority field for its ranking.
The only thing the order field really is good for is ordering your pages in the admin to make lives easier. Nothing more.