Pro tip. Learn (or copy down) your address in Japanese. Or carry it around with you on a piece of paper while you wait for your official documents (resident card, etc.) to come through. Every Japanese furniture and appliance store ships directly to customers' houses.
Groceries: as above, people don't do a week's worth of shopping and throw the mountain of food in the cargo space of the Toyota Suburban. They buy a day or two (or three) worth of food and carry it back with them, be it on a bicycle or on the bus or subway system (if they live that far from the supermarket). They buy what they can carry.
And if you know anything about Japan, you know that there is a conbini (convenience store) on every corner, and the conbini is much better than you can imagine from looking at convenience stores in the States. In the first place, the conbinis in Japan are immaculately clean and safe. There is no atmosphere of repressed criminality, or of an impending armed heist. There are no locked cabinets full of liquor and vape products. The Japanese conbini are bright and full of food and desserts and household items, and now even fresh produce. If you don't want to buy bulky items at the grocery store and carry them home, you can buy them from the nearest conbini (Lawson, Family Mart, Seven Eleven, etc...) which will probably be 200 yards or less from where you live.
Japanese live in rather compact and tidy places, and their shopping habits reflect this. Nobody has huge trucks, and most students don't have cars (or garages) and so when any big purchase is made (like a bed or a fridge) it gets delivered to the address of the student. There are options for you to have delivery made even if you are not at home to receive the package. In the worst case, if you are gone when your package arrives at your door, your delivery will get taken back to the local distribution center, which is probably a mile from your house, and the delivery company leaves a note with the details and the options available to you. You can arrange to have the item delivered later that day or the next day. Its too complicated to write down all the possible options, but suffice to say you needn't be too worried about this stuff - the store clerk should be able to help you out, especially in Yokohama. While they may not speak perfect English, you should be able to communicate well enough.