Detectives believe a white Toyota Hiace was stolen from outside a home on Bouverie Street in Carlton about 10.10pm on April 23.
It's understood an unknown masked man attended a petrol station in the stolen van on Elgar Road in Box Hill about 7.10am the next day.
CCTV footage has been released by police, showing the moment the man filled up the stolen Toyota van with $48 worth of petrol, before entering the store.
Police believe he then selected several items, before driving off without attempting to pay.
The store attendant can be seen in the footage chasing down the van, but he was unable to stop him from driving away.
The same man is believed to have entered a shopping centre on Maribyrnong Road the following day, where police allege he stole an Optus Wi-Fi router and a large battery from a pop-up store.
He is perceived as being Caucasian in appearance, medium build, about 180cm tall, with black hair and in his late 20's.
He was seen wearing a t-shirt with a yellow circular emblem, blue jeans, black cap, black mask and had a black satchel.