That would depend on why they're vegetarian. If they just don't eat meat because they don't like it, or can't digest it, then they might have no problem with killing an animal. It doesn't really matter what anyone thinks somebody should be comfortable with, it matters whether they are comfortable with it or not. If they don't want to do something that's up to them, whether it makes sense to others or not.
The underlying philosophy for becoming vegetarian would need to be explored. Some people are vegetarian due to their religion and if they religion also prescribes being compassionate to all animals, then obviously they wouldn't fish. As Jennifer said if the reasons are more due to taste preferences, then fishing will probably not hold a moral value.
If they were a vegetarian because they didn't want to participate in cruelty to animals, then I would imagine that would rule out fishing as a pastime. If they're a vegetarian for health reasons then I guess not.
Any sentence with a "should" in it implies judgement. Vegetarians are vegetarians for many reasons so to tell them what they "should" be doing makes me uncomfortable. Speaking on behalf of this vegetarian, even when I was not a vegetarian I was not in to blood sports. I have had experience of both sides of the eating paradigm, and I have decided that on either side I have never been comfortable with judgement and universal applications of "shoulds"