"She's playing you"? "Make her more afraid of the feeding tube than of vomiting"? Wow, pretty harsh stuff to say about and inflict on a kid this young.
It is very typical for kids to associate one bad episode with whatever they THINK caused it and then to fear the supposed cause: She vomited after dinner, so she fears food; if she had had diarrhea after eating, say, yogurt, she would fear yogurt would "do that to me again" and would fear it. And so on. It is not rational but it is normal. I don't think she's playing you to get nuts and crackers -- because if she really were playing you she'd ask for something more interesting to eat. And as for scaring her about feeding tubes as someone suggested: Do you want her terrified and resistant next time she has to go to the doctor or the urgent care center? That's what could happen if you start talking blood draws and feeding tubes up the nose to a kid this age. That's what will be in her mind when she next hears the sentence, "You're going to the doctor."
I would go with the person who posted that you should cook a very fragrant meal (but a fairly bland one -- avoid fatty, heavy things like meats, for instance) and invite her to try a tiny portion. I also would no longer make those nuts and crackers available; tell her you ran out and are not getting any more this week, so now she can have bread today, for instance. Don't allow her to say what she'll eat, just quietly make certain things available to her and if she does not eat them, she eventually will. She will not let herself starve.
Don't make a big deal of this or talk about it over and over with her; if she brings it up, gently say, "I'm sorry you got sick, but you're all better now. When you're hungry we have things you can eat." And change the subject. She may be sensing your own anxiety over eating and that is making her even more anxious; it could be that the more you worry and want her to eat, the more she will worry and not want to eat.
Your idea of a little distraction is a good one, too, I think. While she is at the table eating, alk to her about something fun she's going to do right after the meal, but don't talk about the meal itself, or why it's so important that she eat.