Bowser - It really depends. If this is one of the sports or plat formers, he'd kick your dog and burn your groceries. If it's Paper or RPG, he'd start of laughing and mocking you, but end up helping somehow.
Captain Falcon - He'd help, but only to show off how fast he does it. May involve punching targets to desired location.
Diddy Kong - He'd gladly help. But watch out, if you bought bananas, he will steal some.
Donkey Kong - No problems with his assistance, but your dog will fear him.
Falco - He won't help you unless you challenge him about it. Then he has to prove you wrong by beating you at dog walking and grocery carrying.
Fox - He would take some time to help you, if the price was right.
Ganondorf - He'd seal you away in some magic crystal, turn your dog into a demon wizard beast, and eat your groceries.
Ice Climbers - They would gladly help you. Very useful if you live in high altitudes or Alaska. Especially good at keeping your groceries away from wild animals.
Ike - He'd help with nothing witty or comical about it.
Jigglypuff - Gladly help, but would want payback by making you listen to her sing. Would want to sing first. After you fall asleep, will no longer help, and will harm.
King Dedede - Would laugh and mock you, but wind up helping in the end.
Kirby - Gladly help, but screw up. Probably end up eating food and dog.
Link - He would help only if you are aggressive when you ask for his assistance. Otherwise he has demon wizard dogs to stop.
Lucario - No help unless you tell him it will improve his training. May bond with dog.
Lucas - Gladly help.
Luigi - He'd be quick to help, but he'd lose your dog and drop your groceries when a bush startles him.
Mario - Quick to assist you, but you'd end up being kidnapped.
Marth - Not only would he gladly help, he'd do it better than the rest of the cast.
Meta Knight - He would be very hesitant to help. Only if you can convince him it will be honorable to help will he do so.
Mr. Game & Watch - Great assistance, but only if you are traveling in a straight line.
Ness - Eager to help, but you need a lot of room, and your groceries/dog will be burnt when arriving at destination.
Peach - Gladly helps you, but only halfway. Stops when she gets kidnapped.
Pikachu - He would be eager to help you, but you better wear rubber.
Pikmin & Olimar - He'd help, but it would have to be one grocery item at a time.
Pit - Eager to help with nothing comical about it.
Pokemon Trainer - Eager to help with groceries, but he'd randomly attack your dog and throw balls at it.
R.O.B. - Robotic Operating Buddy, he has too.
Samus - Brushes you off.
Zero Suit Samus - Brushes you off and causes you to have to wait a few minutes before going out to walk dog or buy groceries.
Snake - He would help only if you were buying something that was contained in a large cardboard box.
Sonic - He would mock how slowly you are walking your dog or moving groceries, then do it for you to show you how it's done.
Toon Link - He'd gladly help, but something explodes, breaks, or is stolen.
Wario - You think he is helping, but then he just runs off with your stuff, leaving only a smelly trail to follow.
Wolf - He would not help, but he wouldn't harm. Unless you bought Fox's help earlier.
Yoshi - Eagerly helps, but he'll eat the groceries.
Zelda - She'd offer to help, but then disappears for hours/days just before you are going to buy groceries/walk dog.
Shiek - Gives you cryptic advise on how to carry groceries/walk dogs, then disappears.