Hiccups are fairly common among people using the nicotine gum if they chew it "too quickly", but I believe the problem comes from swallowing nicotine that causes the stomach to become irritated and can produce hiccups. If you are getting hiccups from vaping, it could mean that you are using too much nicotine and/or are swallowing some amount of nicotine. The nicotine in e-cigarette vapor is absorbed more slowly than the freebase nicotine in cigarette smoke so if you are used to the speed of delivery from cigarettes you may find yourself taking in more nicotine than you intend...the good news is that this is very self-regulating, once you get enough nicotine to give yourself the hiccups or similar symptoms, most people will stop consuming before they make themselves sick. If you don't stop, you WILL get sick and your body will reject the excess nicotine by inducing vomiting...but most people try to NOT make themselves sick with too much nicotine, yanno?
If it is a problem for you, I would suggest lowering the amount of nicotine you use and/or drinking some water to dilute any nicotine that may be getting into your stomach.