Ghostarchive is a free-to-use archiving website designed to be fast and easy to use.
The process is simple: all one has to do is enter in the link of the page they would like to be archived, and Ghostarchive will store a snapshot of the website as it appeared at the time of archival. The snapshot will include any images and framed content. For some websites, videos are also saved.
Ghostarchive is in the process of adding support for the MementoWeb API. In the mean time, making direct web queries and scraping the result should suffice.