Good Morning Kyle,
EAA Chapters are still unable to own/fly their own aircraft. However, there is a current model that allows chapters to set up flying clubs immediately alongside the chapter to operate the aircraft. The club becomes a separate legal entity and does not use EAA in the title of the name. Being a completely separate entity actually has quite a few advantages. First and foremost is the separation of finances. Chapter members who don't fly the airplane have peace of mind knowing that their dues are directly benefiting programs that they use rather than the airplane. Additionally, this makes financial record keeping much easier. Another benefit is the separation of power. The flying club will have it's own officers and board members, separate from the chapter's. Of course you may get some overlap, but by and large this allows for more efficient decision making with the aircraft and its operation.
Even though the club is separate from the chapter, the flying club members are all chapter members, the aircraft can be a chapter built aircraft, the chapter hangar is used to house the aircraft, etc.
Your question was not completely off course by any means. The reason I asked the above question, was to get a feel for any chapters that might be interested in setting up this flying club model.