Can our members children of our non-profit apply for the foundations scholarship? The process is open to all high school graduating seniors and we have documented proof of profitability,

If it can be shown that their relationship had no influence on their award, yes.

What is the process that the organization uses for deciding who wins the scholarships? If there is a judging panel then no immediate relative of any of the applicants should be on that panel.

Also, I don't understand what you mean by "we have documered proof of profitability" - please clarify what you mean.

Sorry... I was in work mode when typing profitability. We have an application packet that us disbursed to local high schools. We have a grading system for categories such as essay, leadership roles, community service, GPA etc. There is a panel of two people who interview the candidates. The scores are tallied and ranked based on the funds available. We have documented proof of our process and the fairness of the rankings.

I too would suggest that no relatives of applicants be on the panel, but that is weak at best.
I would discuss what might happen to your organization's reputation, fair or not, should a member's child get chosen.
'Documented proof of the process' doesn't discount the fact that part of that process includes an interview and an essay, both of which are subjective.
A parent who wants to make trouble, can. This is why you see so many awards, grants, and scholarships not available to people associated with the giving organization.
I might have the interviews done by, and the essays and interviews judged by, local community leaders from diverse groups, and give them equal weight to the objective parts.
But even that might not be enough to stop one person from causing an uproar.