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 by birthright, 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:
If you are a citizen of the United States, you do not need a passport to enter Canada. However, you should carry proof of your citizenship, such as a birth certificate, certificate of citizenship or naturalization, as well as photo identification. If you are a permanent resident of Canada or the U.S, you should bring your Permanent Resident Card with you.
However, depending on how you're travelling, you may have difficulties re-entering the US without a passport! These are the rules (summarized) for US citizens coming back:
U.S. citizens entering the United States by land or sea are required to present a valid WHTI-compliant document, which include:
U.S. Passports U.S. Passport Cards Enhanced Driver's Licenses Trusted Traveler Cards (Global Entry*, NEXUS, SENTRI, or FAST) Military Identification Cards (for members of the U.S. armed forces on official orders) U.S. Merchant Mariner Document (for U.S. citizens on official maritime business) *The Global Entry (GE) card is only an ENTRY document and may not be used to enter Canada, Mexico or Adjacent Island.
Children: U.S. citizen children ages 15 and under arriving by land or sea from a contiguous territory may present an original or copy of his or her birth certificate (issued by the Vital Records Department in the state where he or she was born), a Consular Report of Birth Abroad, or a Naturalization Certificate. If the child is a newborn and the actual birth certificate has not arrived from the Vital Records Department, we will accept a Hospital issued birth certificate.
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:
Proof of citizenship is required when crossing the international Windsor-Detroit border.
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.