Thanks for the question.
They would be able to join any private Catholic association as long as they are willing to follow and obey the regulations and guidelines of that association. Whether they are baptized or not, would depend on:
- those who spearhead the group, and
- the purpose and mission of the private association.
Although the Church gives its members the right to start their own lay associations under Canons 215, 225 § 1, and 225 § 2, the members of any private Catholic association should:
- take the responsibility for their mission very seriously
- only authorize faithful, practicing Catholics for key positions within the organization, and
- always strive to be in the good graces of their local bishop.
Public Catholic associations are always overseen by the local bishop.
Any and all decisions of any public Catholic association would be guided by:
- the local bishop, or
- whoever he delegates his authority to.
I hope this helps,