I have found:
Ringlink software available at SourceForge.
Ringmaker software at Orca PHP Scripts.
Draupnir Ring Manager for WordPress sites.
Webrings for Writers
I’m not sure how active these are but it gives me hope. All the others I checked which were on web applications were abandoned, dead or not functioning. At least this proves it can still work. Webring.com and Ringsurf haven’t managed to entirely kill every last webring.