What do you do to get rid of gas from your mouth and stomach when siphoning your tank?
If you’ve reached the point where you feel like crap it’s too late to do anything other than drink water and hope you didn’t get enough to actually do your organs permanent harm. If you feel bad, it means it’s already in the bloodstream. It absorbs through skin, which means it absorbs even better through the soft tissues of your mouth, esophagus, etc. Your chance to puke it up was over before you started feeling the effects, has to be damn near immediate because of its rapid absorption rate. If you’d gotten a large enough amount to puke up, you probably would have anyway.
Fuel is caustic, so the official answer is taking something to puke it up is a bad idea as it will cause additional chemical burns coming back up. Call poison control if your don’t believe me, with fuel they’ll say not to vomit.
Get a transfer pump if you’re planning on doing this frequently. A simple fuel bulb works just fine.