I just launched a similar service called https://www.DailySiteSnap.com that screenshots, emails, and archives a specified web site on a daily basis. My use case is to be able to look back at any one day and see what my site looks like, since Archive.org doesn't refresh my page as often as I update it.
Disclaimer: I'm really not trying to over-market myself, but I figured readers of this thread might be interested in my project. Happy to take down this post if it's read as too spammy.
You can always package your site up in a format Archive.org can read:
http://www.archiveteam.org/index.php?title=Wget_with_WARC_ou...
http://www.archiveteam.org/index.php?title=The_WARC_Ecosyste...
Thanks for the info, was unaware of ARC/WARC formats. That said, I still think many people are looking for something simpler/easier, and a daily screenshot is good enough. Particularly, it will guarantee preserved formatting as browsers continue to evolve.
Once you have the WARCs you can upload them to Archive.org and they can be added to the wayback, or you can set up your own service for browsing them, built off something like warc-proxy https://github.com/alard/warc-proxy (Yeah, same name different purpose...)
There is also a MITM version of WARCProxy that will let you store HTTPS sites: https://github.com/odie5533/WarcMITMProxy