Q:

Bentching after vomiting

If someone gets sick during a meal, assuming they are not incapacitated as a result, should they still Bentch, or do we say that they did not gain any nourishment and it's like they didn't eat (or at least like they didn't eat enough to constitute a meal)? 

A:

See Har Tz'vi (OC 1:163) where he discusses this issue.  He first quotes the Panim M'iros (2:27), Chasam Sofer (OC 127), and the Minchas Chinuch (313), who hold that hana'as mei'av is a necessary criterion for birkas hamazon.  Therefore, these opinions hold, if someone ate half of a k'zayis, then vomited it up, then ate another half k'zayis, he would be exempt from birkas hamazon (see also Chullin 103b).  The Har Tz'vi infers from these opinions that if someone ate a k'zayis at one time, there might be grounds to say that birkas hamazon is required.  
However, he then quotes the Gra (Kol Eliyahu, § 9), who writes that if someone vomited out all of what he ate, he is exempt from birkas hamazon because this is similar to a case of delaying bentching until after the food is digested (whereupon one can no longer bentch).  
The responsum concludes by referencing the Imrei Binah (OC, § 14), who writes that even if a person kept a little bit of the k'zayis down, he should not bentch because kdei sviah does not remain in his digestive tract.  
See also Torah Lishmah (§ 125), who writes that someone who vomits, if he won't be able to eat more and keep it down, should bentch without shem u'malchus.