When should my puppy start obedience training, and can they go before getting all vaccinations?

Depends on what kind of obedience training you're talking about, first off. So I will tell you that obedience training that you're going to do at home? As soon as possible and, again, in moderation. You know, with a six week old puppy, you don't put them through bootcamp. Let them be a puppy. If you're going to give them a treat, why not have him on a tabletop like this, hold the treat up high so they're looking at you, and then say, "Sit. Sit."" And watch their rear end drop to the ground. If they're not, hold it up higher. Go above their head so they're going to follow that treat and then they're going to sit. It's a natural response. So that kind of training, heck yeah. I'd do it from day one.
Other small training, like every time I feed my dogs, I put the food down. I make them sit or at least stay and I'll put the food down. I do not let them attack the food bowl. It's by design. I want them to know that they’re going to do what I'm asking them to do. It doesn't hurt. I'm not being mean in any way, shape, or form. But they do kind of learn that they have to do things my way and not their way all the time. And you would be amazed at how many things that spills over to. I mean, keep in mind, I'm a veterinarian. So I see dogs walk into exam rooms all the time. A dog that is well-disciplined even if they don't know 50 tricks, that's fine. But even if they know discipline, they know to sit, they know to stay, they know to heel, those kinds of things—that is such a big advantage for us and for the owners.
It just makes for better dogs when they have discipline like that. So I implore you that even if it's small things, train when you can in moderation. Don't do it for 30 minutes a day. Do it for three or four minutes a day and stop. They’re still puppies. Make it fun. When it's not fun and they're losing interest, stop. Do it again the next day and you'd be much better off for it.

Obedience training kind of varies. Some places have a one-on-one type of obedience training, and other places are more group. I don't usually recommend group classes just because you don't know what other puppies have been exposed to and things until after their complete vaccine series. If you want to go to obedience training, try to stick with one-on-one or have someone come to your home, but puppies definitely need to start learning as early as possible. And they are going to retain a lot of information as early as six weeks old. And so you want to start working with them pretty early on, and they can learn a little bit more, so the earlier, the better for puppies to learn. But we generally start obedience training towards the end of their booster series so that they are a bit more protected and more mature.