Clark County, of which Vancouver is part, has a universal leash and scoop law with a $250 fine. However there are still many opportunities for you and your dog(s) to enjoy the outdoors.
There are two off leash dog parks with a very active non profit society looking to establish more. They are at NE 18th St between 164th Ave and 172nd Avenue (Dakota)and Ross between NE 14th and 18th Streets (Ross). Include map. Visit Dogpaw website for up to date information. www.clarkdogpaw.org
There are many other places where no one is going to complain if you let your pet run free. I live in West Vancouver so will describe the ones I frequent.
The Flushing Channel. The channel was built to divert Columbia River water through the stagnant Vancouver Lake to clean it up after years of industrial discharges. From the parking lot adjacent the Columbia - be sure to lock your car and conceal valuables - walk west on the beach. Your dog will enjoy clean sand, woods and retrieving sticks from the river. Usually there are freighters moored in the river and, in season, you may see thousands of geese and sometimes cranes. Off Leash Dog Map
This is a formal county park with a fee at busy holiday times. If you go there at 7am any day you'll rarely see anybody else. Park in the lot by the caretakers house and walk west on the beach. You will see an Osprey's nest on a log piling as you hit the beach, geese in season and sometimes bald eagles. Dogs love this area, woods to explore, water to chase sticks into and lots to sniff.