Try getting her involved in the preparation of the veggies. At her age, she can stir, measure, break things in half (string beans, etc), help put on spices, help serve. Kids are often much more willing to eat things that they've helped cook.
My nephew had a "veggie chart" in the kitchen. It had every veggie known to man on it, I think! From artichokes to zucchini. My sis would try new veggies as they came into season. Every time my DN tried one ("trying one" meant eating at least half a serving ... putting it in his mouth, saying "yuck" and spitting it out didn't count), they had this really involved rating system that he'd helped devise. The veggie would be rated and placed on the chart, with favorites and "I will never eat this again" preferences noted. For every three veggies he tried, he'd get a $5 gift card to Toys R Us. Since he was saving up for some pretty specific
Lego kits at the time, he had an incentive.
That might not work for a 4YO (DN was, I think, 7 at the time), but something else that makes it a game to try new things might do it. "Let's Only Eat Orange Food" day, for example. Or "Green Day" where you have to wear green clothes, drink green drinks, eat green food, paint green pictures, etc.
And if all else fails, give her a multi-vitamin and wait until she grows out of it. If you, as the parent, set a good example by just regularly eating veggies (without making a big deal out of it), she'll get the point that there's nothing special about them -- it's just food. And she'll eventually get over it.
