It sounds like GERD (acid reflux) to me all of our sons had/have it. Our oldest had it the worst. He had apnea and strider so at 3days old he was put on meds. Our other two we chose not to medicate.
If the baby sounds hoarse after the vomitting or like he's struggling to breathe, he might need to be medicated or seen by an accupuncturist.
My cousin swears by chiropracting, her husabnd is an astounding chiro, but once you start that course of action, you are going to need it for life.
Opting for the meds you can use it for a few months, just to give the baby a break from the vomitting and allow the throat to heal. With meds though the side effects are real and not easy to deal with. But Our son had it so severe the He had to do them side effects and all for four months be fore I weaned him off.
With the accupuncture it allows the body to better regulate the amount of stomach acids. No side effects. Life lond habitual use not necessary either.
You could also get a reflexology book for the library or a borders, etc. I can't remember the exact position, it's something like . . . go to the third rib on the infant's right side, go to the right two fingerwidths, and massage in clockwise circular motion. But I'm a bit rusty on it so look it up first. It helped with our second son.
All three boys of ours hiccup for 24-48hrs straight everytime they have a growth spurt! For them it's become a fun thing to know that when they start hiccuping uncontrolably that in a day or so they'll be taller!
Gerd can be around for life but it doesn't have to be the end of fun.