Hello, my name is***** and I'm happy to help answer your question about Lulu. Based on your description, there are many possible causes for her signs.
Is there any difference in her ability to keep down water vs. food? If so, Lulu could be affected by a condition called Crichopharyngeal Dysphagia or Dyssynchrony. Dysphagia refers to a disorder affecting swallowing. Dogs with this condition will be able to take food and water into their mouths, but when they go to swallow the food or water there is either a functional, anatomical, or timing error and the food/water is not successfully swallowed. They are therefore constantly hungry and desperately want to eat anything in sight, as they are not getting adequate nutrition. Oftentimes dogs with this condition have a harder time drinking water than swallowing hard food.
Another possibility would be a congenital vascular ring anomaly causing a constriction of the esophagus. What this means is that the vessels coming from her heart developed abnormally and are causing a narrowing of the esophagus at the level of the heart. Dogs with this condition usually do ok with liquids but when they are weaned and transitioned to solid food they start showing signs. Signs of this include normal swallowing followed by regurgitation (passive vomiting) of the food. They will often try to eat the food again. Again, these dogs are constantly hungry because food is not passing effectively into their stomachs.
A third possibility would be a foreign body that is blocking the outflow of the stomach. Normally food passes from the mouth, through the esophagus, into the stomach, and into the intestines. If a blockage is present in the stomach we will see them vomit after eating. Usually dogs get pretty sick after living with this condition for a few days, but not in every case.
Lastly Lulu could have an unknown metabolic/systemic disease causing her excessive hunger and frequent vomiting. Examples include a liver shunt, congenital kidney or liver disease, muscle inflammation near her mouth or neck, and others.
The moral of this story is that Lulu should be evaluated by her veterinarian regarding your concerns. The vet will do a complete physical exam and will likely recommend x-rays of the chest and abdomen, as well as blood work to evaluate things like kidney and liver function. They may also want to watch her eat to see how well she swallows, etc.
I hope this answers your question. Good luck to you and Lulu!