I am in Nanaimo and although I don't know about meats and cheese, I contacted the COBBs bakery we have here and they sell fresh bread that you can place into your own reusable bag. I believe there is a COBBs bakery somewhere in Vancouver.
The soap dispensary on main st. is a fantastic place for buying all kinds of things in bulk. You bring your own clean containers, and they will fill it for you. Look up their website for a list of their products... It's quite extensive. You can get meat and cheese in your own container if you go to whole foods. Also, depending on where you live, in Hastings-sunrise, there's a local Italian market that has a deli section with meets, cheese, and olives. I spoke with the manager and they will fill your own container, provided you acknowledge they can't be responsible for the food once it hits your container (and I've never had a problem with the quality of their product). I'm sure there are many more places that will do that, it's just best to ask. Also, with the exception of whole foods, it's best to stay away from the chain grocers, as they are more reluctant to let you use your own containers. Good luck!!
There's the Soap Dispensary, nearby there's Balance Botanicals on Main, Ugo &Joe's on Hastings and Nanaimo (I asked if I could put their cheese and stuff that's under the case in my own container and they seemed happy to do so), the TONS of microbreweries in the East Van area, and Donald's has dry bulk now.
(I don't often leave this neighborhood, so it's the area I'm most familiar with.)
I thought I would update this topic because I have found a bunch of places since I last posted here that are happy to take my jars/bags to refill stuff. I live up at UBC so I mainly stick to the West side, but here goes:
Save-On Foods (UBC): Able to buy dry bulk items in the bulk aisles including snacks, baking supplies, candy, coffee, etc. Also, they are happy to fill my jars with stuff at their meat/fish counter. They don't have a huge selection, but I'm okay with it. They also have a deli section that I haven't tried refilling my jars at, but I don't think it'll be a problem since the other counter does it.
Soap Dispensary (Main St.): I come here for all my household stuff. Cleaning supplies, shampoo, conditioner, stuff to make candles with, etc etc. A lot of personal care stuff here too. ALSO, they've started refilling oils, honey, vinegars, and other condiments here too :)
Jackson's Butcher (West 4th): Anything I want to buy, they'll put in my jar for me and they have a huge selection of meats.
David's Tea (West 4th): No problems whatsoever. Even got a discount for bringing in a jar.
Starbucks: I don't drink Starbucks anymore, but I used to work there so I know for a fact that they can sell you bulk beans and grind them (or not) and put them in your own jar for you.