Question: Can I write a supporting letter for a child who is appealing college dismissal?
It's not clear to me if you're talking about a letter supporting your own child or someone else's (e.g., your employee, a neighbor, former student, etc.). But, in either case, the answer is yes.
If it's your own son or daughter who is appealing a college dismissal, then, ideally, the letter you write will supply specific information that will help the adjudicators to see this child in a positive light. In other words, don't simply provide a list of laudatory adjectives ("hardworking," "compassionate," etc.) but also give examples (recent ones, if possible) to back up your claims.
If appropriate, try to also offer information that your child might not be likely to share that may have spurred an ordinarily law-abiding student to act out. However, if it's information that's potentially mortifying to your child, you should check with him or her first before revealing it. Details about serious homefront problems and parental misdeeds would fall front and center in this category. But, on the other hand, don't try to excuse bad behavior by suggesting that it was spawned by "problems" that are more routine than traumatic.
Yet, often the most effective letters of support come from those with less bias than a parent. This list could include a teacher, college administrator, employer, coach, member of the clergy, etc. Again, specific examples always trump more general lists of positive traits.
If it is your child who is under the gun, and you have not done so already, try offering alternatives to the dismissal, if it looks like the ruling isn't going your way. For instance, how about a semester or year away, either working, volunteering, or attending a different college? How about returning on probation?
Many colleges have zero-tolerance policies in certain areas (e.g., drugs, cheating, violence) that are hard to overturn. But in some situations, there is wiggle room, so a student who seems contrite and eager to take responsibility for the infraction (but not repeat it) may make a successful appeal. It's hard to weigh in effectively without knowing any of the details, but I wish you well as you proceed.