Sorry @AaronI've got two sick kids right now, didn't get to this last night. There's probably a way of getting a rental scooter or bike but I would just be googling those so you can do so just as well. Evo mentioned above is one way car share that will get you to some but not all places (city of Vancouver and North Van only) without just hanging onto the car. The good news is that transit is quite good here, but not to all the places you'd be looking at going. More on that later.
Best news here is that Jericho is a great hotspot, with ocean/forest/meadow/pond habitats in it. Migrants here seem to follow the coastline so it's a pretty good spot. You'll get Anna's and Bushtits down the street from wherever you're staying in Kits, and Jericho should get you Bewick's Wren. It's as good a place as any to try for the warbler although I don't know about seasonality on that, and a good place for Short-billed Gull too although it's a bit early for them to be here in any numbers. If that's walkable from home it's a great starting point as really it's a nice place to go through casually. Also be sure to bring carrots to feed the bunnies in the park.
Stanley Park is also great. It's probably the best place to look for sapsuckers without going over to North Vancouver. I've seen them a bunch in trees around the rhododendrons beside the golf course, by second beach. Walking along the trail there to/from lost lagoon is nice, and the forest trails have lots of stuff in them. Getting out onto the seawall around the point gives you good views out into the ocean where there can be rafts of birds, but these are again quite seasonal and I don't know how good July will be. There are lots of places to rent bikes/ebikes etc just outside the park that you could use to get around if you wanted to.
I'd suggest looking at Maplewood Flats hotspot in North Van. It's a really good place to go, generally, and they have nesting boxes and a good population of Purple Martins. You can take an Evo to close to there, but you may have to walk a ways to find one for a return trip. It's about 90min to get there on transit.
For most of the aquatic stuff you're going to need to get down to either Richmond, Delta, or Boundary Bay. Of these the easiest without transport is Iona Beach. If you can arrange a bike or a scooter this is really easy: get to the SkyTrain, go to Templeton station, and it's 5km to the park directly from there. The basic transit math is that buses within the city of Vancouver are frequent, SkyTrain is frequent, buses down anywhere interesting from the SkyTrain are very infrequent. So Iona is good because it's easily accessible without that second bus. There are other Richmond hotspots that will be similar but that seems to be a key one.
If you end up having access to a vehicle for a day I'd suggest looking at Colony Farm, and just generally looking at the hotspots along boundary bay as I'm not sure which are the key ones. Maybe this one is a good start. Roberts Bank and Tswassen ferry terminal jetty are others to look at, as well as Reifel Bird Sanctuary (which requires prebooking up to three days in advance).
TL;DR Jericho is great, let me know if you have specific questions. This is probably a good resource for much of this as well. https://birding.bc.ca/regions/vancouver/