While I don't know the best method, I do know that there are tons of ways to actually go about getting the Grand Company Seals. They are as follows:
Grand Company Hunting Logs -
FATEs - Higher level FATEs give more seals.
Roulettes -
Grand Company Leves -
I don't yet know personally if there are other ways. If there are that anyone knows of, feel free to add them in a reply and as time goes on I will gladly work on editting this post.