The files are only available after submitting the form, therefore I'm not sure if we can include a downloader in MNE. Furthermore, the website doesn't specify the license of the data files, so I'd be careful.
However, many BCI data sets (including some from the BCI Competition IV) are available at http://bnci-horizon-2020.eu/database/data-sets, licensed under CC. I think a downloader for these data sets would be very useful. Do you have time to contribute a PR?