The reason I ask about the constipation is that constipation isn't seen commonly in young dogs, and it would be much more common to see straining due to large bowel diarrhea. Sometimes after having diarrhea, when the bowels are empty, dogs will strain as though having a bowel movement and nothing comes out, so if we don't realize they had been having diarrhea we may think they are constipated. Just trying to figure out if that is the case here!
If he is otherwise alert/energetic, his gums are pink, he isn't painful in the abdomen, and is breathing normally, you can try treating him at home overnight for vomiting by fasting him for 6 to 8 hours. This means nothing to eat or drink until he hasn't vomited for 6 to 8 hours. If he has to go outside take him on a leash so he can't eat grass while outside. We need ot give his tummy some time to rest. If he doesn't vomit, then you can start by offering him small amounts of liquid, such as water or preferably pedialyte every few hours. If he goes another few hours without vomiting, you can start a very small amount of white rice and boiled chicken breast. Here is an article describing home care for vomiting and diarrhea:
The concern with your pup also is that he is so young, it is very easy for him to get dehydrated quickly, so we don't want to let the vomiting go on for too long without seeing the vet. The other thing I worry about in puppies with vomiting and possible diarrhea is parvo virus, which can cause vomiting and diarrhea and needs to be treated by a vet. If your pup is still vomiting later tonight despite the fasting, then he should be seen ASAP. If he is lethargic/weak, has blood in his stool, has pale gums, has a painful belly, then he needs to be seen ASAP.
I hope this information is helpful to you. Please let me know if you have other questions. If you do not, please take a moment to rate our conversation so we can know if you were happy with your service. Thank you and good luck!