The National Archives has a system for veterans or their next of kin to order their service information at http://www.archives.gov/veterans/military-service-records/. The site indicates that:
Use our system to create a customized order form to request information from your, or your relative's, military personnel records. You may use this system if you are:
A military veteran, or Next of kin of a deceased, former member of the military The next of kin can be any of the following: surviving spouse that has not remarried, father, mother, son, daughter, sister, or brother.
The directions for ordering the records if you qualify to request them is:
How to Initiate a Request for Military Personnel Records:
Click on the "Request Military Records" button below to start. This will launch a separate window.
Enter the required information in the system to create your customized request form. There are 4 steps that you need to navigate. The system will guide you through the steps and tell you exactly which step you are on.
Print, sign and date the signature verification area of your customized form. If you don't have a printer, have a pen and paper handy and we will guide you through the process. This is important because the Privacy Act of 1974 (5 U.S.C. 552a) requires that all requests for records and information be submitted in writing. Each request must be signed and dated by the veteran or next of kin.
Mail or fax your signature verification form to us, and we will process your request. You must do this within the first 20 days of entering your request, or your request will be removed from our system.
I did all that, printed out my signature verification form, addressed an envelope and sent it off today. Pretty neat - the cost is basically one stamp.
If you are not the veteran or next-of-kin, you can still obtain information about a service member by filling out Standard Form SF 180. More information about this is here.
I'm surprised that the system doesn't allow grandchildren to obtain these records. It makes it critical that children of veterans obtain these records before the children die. If your ancestor served in this war, and you haven't ordered these records yet, go for it!