Upvote:3
How much stuff do you have?
You will not be able to move personal effects (term customs uses for your stuff) between countries without a residence visa. If you show up at the Peace Bridge border crossing with a rented truck they will deny you entry as it looks like you are moving down there (which you are). Note that the rented truck is no problem at all, but you will have to drive it back to Canada.
If you have a lot to move you can use a freight company (UPS is a parcel delivery service). The container will simply be stored at a bonded warehouse until you arrive and clear it through customs. Not worth it unless you can justify a half-container.
The most cost-effective way will be waiting until your US visa comes through. If you need to move out now, find a self-storage in St. Catherines (there's lots). Rent a truck and drive down there. Put everything in self-storage and return the truck to Toronto.
After you get a place in NY, rent a truck and drive up to the self-storage. Deal with US customs, drive to NYC, unpack and return the truck.
Canada won't care (or even check) on the way out, and an empty truck coming in is no problem, especially with a new US visa and a self-storage receipt. You will need to tell the rental company you are picking up stuff in Canada as there is extra paperwork they need to produce, but this happens all the time and is no problem at all.