Your den won't have to create/hide a geocache for the Webelos Camper requirement. Just go and find one. But the Boy Scout merit badge for geocaching requires the scout to "plan a geohunt for a youth group" and "set up a course with at least four waypoints".

I understand this. In my area, Geocachers have a bad reputation of messing with letterboxes and playing pranks on letterboxers as well. I would prefer to plant a hybrid as a way to get as little into geocaching as humanly possible when it comes time to tackle this requirement with my den.

Is it required to exclusively use the geocaching.com website or can any geocaching listing service be used? There are (non-commercial) alternatives! OpenCaching North America (www.opencaching.us; "OCNA") is one such alternative. It's a completely free service with no restrictions on access. The service is run by a small group of volunteers and they are considerably flexible in what is allowed when compared to geocaching.com. I'm pretty certain they will allow an all-clue-based Letterboxing Hybrid cache as long as there are coordinates for a starting location.
Anyone can view the site and listings and can download the caches without creating an account. That may be good if you want to make the information available to others who otherwise won't bother using the website.
The flip side is that most geocachers can't be bothered using more that the one site, so there are far fewer users on the alternative sites. In a couple years, my 65+ OCNA hides have netted less than 10 finds in total.
FWIW, OCNA is part of the larger "OpenCaching" network of sites; there are similar sites for Australia, UK, Germany, Italy, Spain, France, Netherlands.... OCNA allows cache types that are "grandfathered" on geocaching.com - Virtual, Webcam, and Moving caches. they also have unique cache types such as Guestbook caches, MP3 caches, BIT caches and (coming soon) Benchmarks