Do you have your DD214 or other separation documents?
I do.
You will need your social security number, your date and place of birth, your military status information such as discharge type, service dates and your exit rank as well as your DD214.
Does that information include burial in a tribal veterans cemetery?
This information is for VA national cemeteries.  The other cemeteries provide their own information.
How would I apply by mail?
You would need to fill out an Application for Pre-need Determination of Eligibility for Burial in a VA National Cemetery VA Form 40-10007 and sign it.  You would sent that completed application with copies of your supporting documents to the National Cemetery Scheduling Office in St. Louis.
Can my sister who is 17 and currently serving now apply for burial using the same method you are telling me?
If a person is under 18, mentally unable to make decisions for themselves, or physically unable to sign the form they can have a court-appointed representative to sign the form for them.  However, if your sister is active, she is not a veteran. She does not yet know the terms of her discharge.
Does this also pertain to my own adult children?