For a period of time in the early 2000s, Dr. Demento allowed paid access to his shows through his own website.
Still appears to be the case. I didn't try to purchase any shows, but his website still shows them as available.
More than "for a period of time." Demento
never stopped doing his weekly shows after leaving terrestrial radio. He simply switched directly to doing them online at
drdemento.com, and he has continued doing them there each week right up to today. The archives K.M.R. says appear to still be available not only are still available, they're actively maintained. They contain not only every show Demento has done since moving online, but also full untelescoped airchecks of the majority of shows he did during his terrestrial radio timeline. New archival tapes of those broadcasts continue being digitized and added to the archives weekly. And those archival tapes consist not only of the syndicated versions of his broadcasts, but of the special
live KMET / KLSX / KSCA versions he did just for Los Angeles listeners (which tended to have slightly customized playlists versus their syndicated counterparts -- customized to eliminate a certain percentage of the tacky, cornball stuff in favor of more refined and/or risque things).
The unabridged playlists for every broadcast ever done, from Demento's roots through today, can also be seen (and searched) at
dmdb.org, which is an independently-run fan site whose owner knows Barry and gets each week's playlist info directly from him. The site's owner has been running it for decades -- it's very retro HTML 1.0, and he's also done tons of his own past research crowdsourcing additional playlist data that Demento himself no longer had from fans with their own airchecks.
All totaled, between those two sites, you can pretty much revisit everything Dr. Demento has ever done.
Edit:
* Direct link to his complete playlist history:
https://dmdb.org/playlists/
* The official list of his remaining shows and their topics, before he retires this October:
https://dmdb.org/playlists/newtopics.html