A container on private property, such as a driveway or a fenced job site, needs no street permit in Vancouver. The permit question starts once any part of it sits in the public right-of-way.
For that, the City requires a street use permit. Public Works lists "Large Garbage/Recycling Container or Drop Box" among the short-term activities that need one, alongside portable storage units and temporary storage of equipment. Reviews look at safety hazards, traffic congestion, and liability, and most projects in the right-of-way also need a traffic control plan, which the City reviews free of charge. Call Traffic Engineering at 360-487-7729 with questions about a specific spot.
There is a second rule here that matters more than the permit, and it is unusual. The City of Vancouver contracts exclusively with a single provider for waste collection and hauling inside the city limits, and the City states that the arrangement covers construction activities and demolition waste from job sites. The City publishes a short set of written exceptions, which cover a business hauling its own material in its own vehicle and a subcontractor moving debris in a truck already working on the site.
Separated recyclable material follows a different track. For that, the City requires a commercial recycler it has registered, and it publishes the current list of those companies on the same page.
None of this stops you planning the job, but it does decide how the container is arranged. Give your account manager your exact address and a description of the debris, and they will confirm which of the City's rules apply before any date is booked.
Before you book, confirm with the city that a container can be placed for your job at your address. Rules differ from city to city and they change. Your Prime Dumpster account manager will help you check, and will tell you plainly what your options are here.