Can I download these shows to my local device?

Yes, you can download the TV shows from the Internet Archive.

Just click the down arrow on the right side of a show title button. This will show the Additional Actions Menu. If you click on "Download files", it will open the Internet Archive site in a new tab or window which displays the available file formats for download.

While a video is playing, just scroll down on the page containing the video and you'll see a section which says "Download Options" (see below). Select the video file formats you want and download the files directly from the Internet Archive.