You have to fast before surgery so even if you do throw up, nothing or nothing much should come out. I threw up twice at the hospital but it was mostly dry heaving. Some stuff came up but not a significant amount and no mess was made because the nurses were always at the ready with barf bags. I used to be emetophobic too and I still hate to puke, but you know, it's never as bad as you build it up to be, especially because when you're done you actually feel great as opposed to just normal.
Keep in mind too that you will be in a hospital and you will be hooked up to an IV the entire time, and if you tell a nurse you feel nauseous, they can send medication straight into your system and it will be very fast-acting. When i initally woke up out of anaesthesia I mentioned feeling sick, they put something in my IV and the feeling passed right away.
Also, your surgeon might prescribe you anti-nausea pills for home use (mine did) so you can continue to medicate for it once you're out of there