You can begin this side quest when you reach Lohgrin, First you need to make all shops on every city have 3 stars or above. After all of them reach lvl 3 or above, simply talk to the man beside the shop who will tell you about the Mythril. He will give you Tough Axe.
Now go to Aroundight Forest, and then go to “!” location on your map. Examine the stone and you will get the Myhtril. Now go back to Lohgrin and talk again to the man who gave you the sidequest. You can buy Mythril equipment now.