How do you contact a soldier on active duty?

Find their Facebook page. If you're an immediate family member contacting them about an emergency, you can use the Red Cross to do this - in fact, in order for a service member to be able to take emergency leave over a family matter, this is how it must be done.

There may be one other option, if you know both their first and last name, and it's not a name so common that they have to add additional characters to distinguish them from other service members with the same name, and that's the Army Knowledge Online (AKO) system. You could try emailing them at (first name).(last name)@us.army.mil. Otherwise, there are no resources available to enable the public to track down and contact service members.