YC has a long tradition of soliciting applications in particular areas—see the links below. Of course, founders with other ideas are entirely welcome to apply.
Not sure if the Wayback Machine supports API-based on-demand archival, but they do partner with firms like Cloudflare to auto archive websites: https://archive.is/O9SAO
There's availability API but it won't tell us if the archived webpage does indeed display content (say, if it was behind a paywall or cookie banner): https://archive.org/help/wayback_api.php
Startup Ideas We'd Like to Fund - https://news.ycombinator.com/item?id=250704 - July 2008 (380 comments)
A New Experiment: The RFS - https://news.ycombinator.com/item?id=766703 - Aug 2009 (57 comments)
YCRFS 1: The Future of Journalism - https://news.ycombinator.com/item?id=766796 - Aug 2009 (56 comments)
YCRFS 2: New Paths Through Product Space - https://news.ycombinator.com/item?id=768425 - Aug 2009 (21 comments)
YCRFS 3: Things Built on Twitter - https://news.ycombinator.com/item?id=904193 - Oct 2009 (67 comments)
YCRFS 4: New Ways to Use Live Video - https://news.ycombinator.com/item?id=904195 - Oct 2009 (19 comments)
RFS 5: Development on Handhelds - https://news.ycombinator.com/item?id=950781 - Nov 2009 (44 comments)
YCRFS 6: iPad Applications - https://news.ycombinator.com/item?id=1160140 - March 2010 (106 comments)
YCRFS 7: Applications of Facebook Instant Personalization - https://news.ycombinator.com/item?id=1638004 - Aug 2010 (29 comments)
Request for Startups: Kill Hollywood. - https://news.ycombinator.com/item?id=3491542 - Jan 2012 (664 comments)
New Requests for Startups - https://news.ycombinator.com/item?id=8308666 - Sept 2014 (390 comments)
Requests for Startups Refreshed - https://news.ycombinator.com/item?id=12592619 - Sept 2016 (124 comments)
Request for Startups: Water - https://news.ycombinator.com/item?id=13311257 - Jan 2017 (224 comments)