There is no requirement for any particular corporate or tax structure for a flying club.
It *is* worth asking what that structure is and whether they pay taxes, and if you're going to be an equity member I would say it's a requirement. I know of one club that doesn't bother to file their taxes, and they're gonna get shut down someday when the IRS figures that out and takes their planes to pay all the penalties.
The equity club that I was involved in managing was a for-profit C corp. Nothing wrong with that, and it was actually financially advantageous to us during the time I was involved. Our plan was that if that situation changed to go for (c)(7) status but that was looking VERY far off.
Nothing sounds dubious in what you've posted,
@Scrabo - Not saying it is a good situation, but nothing that you've specifically posted should scare her away. Her exposure, if it's not an equity club, is zero. Her exposure if it is an equity club that's a corporation would be limited to the amount of money she pays for the share. Her exposure if she decides to become the treasurer and they lack D&O insurance could be higher, but other than that she shouldn't have anything to worry about.