I think the NCAA stipulates that IF your "home conference" sponsors football, you must play in that conference
(I think.... Not sure how this explains Notre Dame remaining independent, while a member of the ACC for all other sports, unless they finagled the rules to be an affiliate in all other sports - don't think this is the case).
Army / Navy don't have to play FCS in the Patriot, since they choose to play FBS, and the Patriot sponsors only FCS.
So certainly, if a cluster of schools from conference(s) that currently do NOT sponsor either FCS or FBS wanted to form a FB conference, I think they could.
However, to be considered an official conference, there may be a requirement of minimum of 6 or 8 members (don't recall - I think the WAC may have had the number lowered OR got an exemption while their FB league was disintegrating...)
CAA, MVFC, Pioneer FB leagues are examples of the concept ou are talking about.
The only conference sport that is required to be played in a school's "primary" conference is basketball, because by definition, a school's conference membership for basketball determines its primary conference. Any other rules about where a school plays its sports are set at the conference level. Since the NCAA requires a conference to sponsor multiple sports with a minimum number of members (including a certain number of "team" sports) to receive voting status, generally speaking, a school will play nearly all of its sports in the same conference.
The WAC sponsored legislation that lowered the number of members in a conference to seven "active" Division I members (i.e. members not in transition from Division II). All seven of those must play men's and women's basketball in the conference. The conference must sponsor a minimum of 12 sports (six for each gender), including two other women's team sports (volleyball, softball, soccer, lacrosse). The conference must sponsor either two other men's team sports, or football. One of those three team sports for each gender is allowed to only have four "full" members from the conference and two affiliates to obtain the minimum of six schools. FBS football, of course, has a higher minimum.