How Do I Have Other Students Disclose Their Disability When Running A Student Organization?

Hi all, thank you in advance for the advice. I am really at a loss here, and maybe this is a dumb question and the answer is very simple. However, I want to do this right the first time.

So for context:

Location: Michigan

Goal: I just want to know if there is a way my organization can legally and discretely ask- and have students comply with us asking- for disclosure of a disability that may prevent them from participating or even showing up to rehearsal

I am a student enrolled at a public university in Michigan. I am also the President of an A Cappella group on campus. This group is not university sponsored- meaning that we are completely student ran, and organized. The university policies are not enforced, nor apply when it comes to this organization. Obviously, the Law applies to everyone, and basic human rights are always followed no matter what, but when it comes to disability services...I am having a hard time navigating what to do.

This past semester we had a situation with a member of our group who has been with us since the beginning. She has had attendance issues, to combat this we have implemented policies that we have specifically restricted wayyyy down to make sure she attends and follows our rules and is overall a team player. Well, this past fall semester she missed a rehearsal and technically broke policy. When we went to send her an email and follow through with the disciplinary action of our policy, she stated: "I had a P.O.T.S flair up, since I have POTS, and I was asleep all day...I am so so sorry, I didn't mean to, but I didn't have any control." Our board had a full blown discussion on how to handle the situation because we realized... we don't have a disability statement or policy. Obviously, we cannot punish someone with a disability for something out of their control. Now, I have no idea how to approach this. This student has been with the group for so long and hasn't said anything about her disability until just recently, not that she should of had to, but it could have prevented the confusion ( and us thinking she's just making another excuse) if we have some form of evidence of a disability or learning need/accommodation?

I don't know. I am look for advice on what kind of policy I should implement, and how to even go about it. Can I- another student- ask for a disability statement? As a leader of a club, I would like to, but legally I don't think I can. We obviously do NOT discriminate, and we want to prevent situation like this in the future. I don't want to punish someone, under one assumption, when there's an underlying disability in play. What do I do? We have a constitution, an bi-laws that say that we do not discriminate, but how do we handle disabilities we cannot see on the outside?

Or...do we just hope that members tell the truth? Is there something I CAN do? Or is this a gray area I should let go? I just want to treat everyone with justice, and equality. Please help.

Info:

We have thought about following university policy, but have a hard time asking for paperwork. We do have a faculty advisor attached to our organization as an overseer. We do have a student organization committee on campus to oversee all student organizations are registered but seen as a non-attached organization/non-profit. If that makes sense? We as a group have not ever asked students to provide paperwork, or disclose disabilities in the past. Our university policy consists of a discrete email from disability services to the professor of said class where an accommodation is made. I just want to know if there is a way my organization can legally and discretely ask and have students comply with us asking for disclosure of a disability that may prevent them from participating or even showing up to rehearsal.