Hello,
Yes, vaccines can cause diarrhea. However, for an 8 week old dog, diarrhea is also commonly caused by intestinal parasitism. I do not necessarily mean worms. Parasites can very frequently include the microscopic bugs like giardia and coccidia. So, I would strongly encourage a fecal (poop) test. Bring a fresh sample of feces to the vet and request a fecal test be sent "OUT" to the laboratory (not evaluated in the hospital, that is less accurate).
Even if the diarrhea resolves, I would still strongly encourage a fecal test for every single puppy that is examined. If you observe any vomiting, hives-like reaction on the body, or facial swelling, this would be consistent with a vaccine reaction. However, this would usually occur within the first 24 hours following vaccination. Hope that helps.
Please let me know if I can be of any further assistance.

Yes, vaccines could certainly be responsible for some transient diarrhea. Remember, dewormers are for just that, only worms. Every single puppy and kitten I see MUST (at least I strongly encourage) have a fecal test performed. Giardia and coccidia are VERY common microscopic intestinal parasites that must be evaluated for and treated. I strongly encourage the veterinary examination and fecal test. In addition, diarrhea can be dangerous in a puppy, since it can predispose to quick dehydration.