Mythril weapons can be acquired three ways (without cheating). Trading, Trade Items, and reward from random clan engagements.

As soon as Nono opens Trade items, go buy from him. He always has the same items listed, and the "Metal Sword" is a Mythril Sword. Any item listed as "Metal -weapon type-" in future trades should be a Mythril weapon, so be on the lookout every time you link with someone for new trade items.

Random clan engagements give them to you as random rewards. There is no known way to increase the rate at which you get them, or to acquire a particular one (not doing missions doesn't affect the rate either).

Oh, and no enemies ever wield them, so Steal: Weapon won't help you.