How do you even know it's a purebred Chihuahua? At 3 weeks old, you can't tell--unless you know the parents.
Contact the Humane Society and ask them to find a foster home for the puppy. If you want to adopt him, you can do so when he's old enough. Besides, even if someone did put him in your yard to die in a shoebox, you have no way of knowing it that was done by the owner. Could have been somebody malicious, like an ex-boyfriend. By going through the Humane Society, the owner will have a set period (usually 3 to 7 days) to claim the pup--if that doesn't happen, then later you can obtain *legal* ownership of the pup. If you just keep it, the owner can show up at any time and reclaim the dog and there won't be a thing you can do about it. His property.
And if you try to raise such a fragile puppy yourself, you probably have about a 50% chance of success. Do you want to be responsible for killing this puppy? Does his/her life mean so little to you that you wouldn't get it expert care?
As for the person who claims to be experienced, obviously she doesn't know as much as she thinks she does: "I would vaccinate at 4-6 weeks if it were me since the pup isn't getting moms milk". No vet will vaccinate a 4 week old pup because the vaccine won't take due to interference by maternal antibodies. Even at 6 weeks there is only a 25% chance the shot will be useful. Most vets won't give shots until 8 weeks of age (even then, they're only 40% effective--which is why you need a series of shots as the pup gets older).
And whether or not the pup continues to get his mother's milk make NO DIFFERENCE regarding antibodies. Antibodies are transferred with colostrum milk during ONLY the first 12 to 24 hours of the pup's life. After that, they are no longer absorbed directly through the pup's intestines but continue through to be digested along with the milk.