A slow feed bowl is required - Skid Stop Slow Feed Bowl.
You do not tell us what breed you have but you need to match the puppy kibble with the size of pup. Small kibble for small breeds etc. Do not feed small kibble to a large breed pup. They can vacuum the stuff like magic.
I would also follow your vets advice and do not leave food out. Pick up what is not eaten and feed twice a day only.
Do you have other dogs being fed at the same time? If so the pup may think it needs to get it's share fast or lose out. Ensure you have good seperation and do not let a dog that is finished eating wander near the pup until it has also finished.
The vomiting is usually a short term issue and should stop as soon as the pups metabolism balances out.
Border River Pet Resort