My US passport took too long to be shipped to me, and it will arrive after I need to leave. Will I be able to cross the border from the United States to Canada with a birth certificate (copy) and a passport locator number?
If you are a US citizen, you can indeed cross into Canada using your birth certificate and a photo ID. For safety reasons, you may want to bring the original birth certificate, or at least a certified copy. If you are not a US (or Canadian) citizen or permanent resident, you cannot enter Canada without passport:
However, depending on how you're travelling, you may have difficulties re-entering the US without a passport! These are the rules for US citizens coming back:
So if your passport is delayed, make sure it arrives at your place in Canada on time.
While Jonas' answer is indeed the legal situation but note: there is no penalty for breaking this particular law and more importantly, no US citizen can be denied entry to the USA if said person can prove citizenship. This is more generic than the USA, this is one of the fundamental meanings of citizenship. A birth certificate is a good start, the more IDs you have, the better. Expect a long, painful process border crossing if you do it without a passport.
It's a whole another topic whether an airline would let you board on the grounds of "Oh, I will convince the border guards, no worries". You might need to fly to Toronto or Vancouver and get across on land. If you don't drive, then the easiest is probably the Windsor Tunnel Bus to Detroit. From Toronto, there's a VIA Rail three times a day to Windsor. There are no similar services in Vancouver, you'd need to walk a lot. Note this bolded note on the Tunnel Bus page:
Every other border crossing carrier I am aware of, BoltBus, Greyhound, Amtrak, Quickshuttle all wants a passport. Let's see Crossing from to Canada to the USA w/o passport on public transit? what else is possible.