I thought that you could show pups in conformation between 3-6 months? Or is that only at matches?
Here in NE Ohio and NW PA, do not bring unentered dogs to a show. You will likely be kicked out. You can bring them to matches and no problem. That is probably what that six month rule is all about. You cannot enter a dog in obedience, rally, or even CGC that isn't six months old, so it is easy for stewards to see a young puppy and tell the owners to get it out of there.
Sorry, but if you get shots starting at six weeks, you may be finished with vaccinations including rabies at 4 or 4 1/2 months old, but that doesn't mean there aren't other diseases with no vaccination connected to it on the grounds and pups and elderly dogs are the most likely to succomb to such diseases. So it is really a safer practice to allow your pup to grow up a bit before exposing it to the atmosphere.
The matches are where you go to socialize, for fun, to take pups (carefully and within reason), etc. The shows are for dogs that have registered in advance and paid up front -- a hefty price tag too nowadays. $29/entry, uhg! on top of gas, etc.