For the vomiting
The neosporin shouldn't have hurt him but if you put a lot on there then the fats in the petroleum jelly might be causing an issue.
An over the counter med you might try to see if that calms the stomach down would be Pepcid or Tagamet or Zantac.
Be sure to tell the vet when you see them if you use any of these.
You can read about dose amounts (and cautions) scrolling down on the pages here
But if he is vomiting a lot, then if you can see a vet with him tonight you should try to do so. Dogs that are vomiting can end up dehydrated pretty quickly. You can check for dehydration by pulling the skin gently up over his shoulders in a 'tent' and release it. If it goes right back down that is good but if some stays stuck up that’s a dehydration indicator.
For the paw licking issue
This is a common symptom of allergies in dogs.
Paw licking and skin itching is often a sign of allergies. You may see red staining to the fur in the licked areas or raw spots where the dog chews itself.
The skin itches from allergies, the dog licks and chews often trying to soothe the itch, making the skin raw and allowing other infections to set in which may itch even more.
You can read about allergies and dermatitis in dogs here
Dogs can develop allergies to foods, and to inhaled items, and contact allergens such as rug cleaners or chemicals including lawn chemicals or even flea bites.
You might want to try a different dog food that has no ingredients the same as what you feed now. Wellness Fish and Potato works for many dogs for example.
You might want to see if some plain Benadryl helps with the itching. A usual dose is 1mg per pound of dog.
Or you may want to consult with your vet and consider doing allergy testing.
If your dog is not on a flea prevention that might also help.
Hope this helps you!