Having had a quick look at this for you, there's not a lot of low fare airline service on the routes that you're looking at.
Currently there are flights between LAX and both YVR and YYC by Air Canada and Alaska. (Each of them carries a number of other airlines' codes, including MX, NW, NZ, QF and UA - don't be confused by that, as those airlines don't actually fly the route.)
To YVR there is also
HMY Airways, which appears to be a low fare airline but whose website intriguingly allows booking only ex-Canada, although you can do both one ways and returns. You might have to ask them whether you can start a trip from LAX.
To get from YVR to NYC airports there are flights by AC, CO and UA. There don't seem to be any non-stops from YYC to NYC, so you would have to connect somewhere.
If you have to connect somewhere on any of these journeys, it is possible that you could connect between low fare airline flights (online or interline), but you would have to do a lot of searching of potential connection points and the airlines which service those points to find out. Even if it were technically possible, it may be that building the fare out of two separate trips would mean that it's cheaper to fly on a major anyway. (And - always - it may be cheaper to fly on a major; low fare airlines are not necessarily cheaper.)
Virgin US (whatever they may eventually be called) is the only low fare airline of any size that I know has plans to start up, but I would die of astonishment if they were in the air by June.
Hint: If you want to find out which airlines serve a particular route, one way of doing this is to go to the website of one of the airports and look at their flight arrivals/departure information.