He can be found outside the theatre entrance where SWEETHEART'S QUEST FOR HEARTS takes place. He'll state he's lost his most important SWEETHEART item and requests the group find it. Go down the set of stairs to the floor MARI is on, to her right a shady SPROUT MOLE with a dialogue prompt should be waiting, interact with him and he says he'll help if you give him the MAYOR MOLE'S B.E.D. This quest is one of the two subquests of the PESSI'S THING quest alongside SWEATY THING.
MAYOR MOLE can be found by going down the ladder on the left of the SHADY MOLE (and the right of MARI), and then taking another ladder down on the right of that. MAYOR MOLE needs you to find his son so go back up the ladder to the previous floor and down the one to the left of the SPROUT MOLE shaped house. On this floor, at the far right, the SPORT MOLE should have a dialogue prompt. Give him 1 TOFU and he'll provide his most precious item, a BIG AIRHORN.
TOFU can be obtained from TOFU JASH two floors up for 1 CLAM at his stall (head up the ascending ladder also on the left of the SPROUT MOLE shaped house). Head down the ladder to the left of SPORT MOLE to see a sleeping SPROUT MOLE in a B.E.D. Use the BIG AIRHORN to wake him up. The mole will reveal he's the mayor's son and the B.E.D caused him to oversleep, he'll rush out of the room with the B.E.D.
Return to the MAYOR MOLE and he will declare his B.E.D invention too powerful as it's ability to make others oversleep is counter-productive to a good nights sleep. He'll entrust the invention to you.
Upon giving SHADY MOLE the B.E.D., the quest will be completed and he will then reward the group with BLACKMAIL, giving them hints on how to use it on ROSA.