Let's say I start a fresh new private clan. I would only use that for Clan Avatar purposes. What are the minimal requirements for me to summon the Clan Avatar and get the 6% skilling bonus? Are there any clan resources/tier that would be required?

You need a citadel, first of all.  That requires five people to simultaneously stand on the stones to start the ritual and open it.

 

You may have one avatar with a tier 1 citadel, that is, just starting out.  However, you have to build the habitat first!  To do this, you'll need 3,000 timber.  A second/third avatar requires tiers 4/7, respectively, along with more materials.  You also need to assign any eligible people at least admin rank and the 'Avatar Warden' under job title, or they won't be able to summon it.  This includes yourself.

 

Citadels also require 5 people to visit each week or they deprecate and lock out anyone from even entering them until 5 clanmates successfully visit within the same week.  I'm not sure if the habitat would get removed but it's a moot point considering you wouldn't even have access to it.

 

So to summarize, you need 5 members at least contributing enough as a net to meet weekly upkeep to benefit from the privileges of a clan avatar.

^ Plus you need to pay 300 anagoric orts a week to receive the buff of the avatar.

So can I just create 5 free accounts and complete/cap each week so that would be fine?

So can I just create 5 free accounts and complete/cap each week so that would be fine?

The citadel is a P2P feature.  F2P accounts aren't able to visit or contribute.