We also generally steer clear of the iron-enriched cereals; not because we're avoiding grains, but I just don't like giving her super processed foods. (Oh, well, except Cheerios. She's addicted.)
We're vegetarian, and DD's ped told me at her last visit that we
had to get DD on an iron supplement because "she's missing a food group."

We're changing pediatricians.
We don't give DD any supplements; I think her system is too little to handle them well, she's still getting most of her nutrition from breastmilk and is therefore probably just fine on most of her nutrient levels, AND iron in particular can have some pretty bad side effects.
There are plenty of vegetarian foods that are really good for babies that are high in iron: spinach, apricots, prunes, sweet potatoes, butternut squash, brown rice. You can cook any of these and make into a puree. One of DD's favorite foods is an apricot puree that I make - I just soak dried apricots in boiling water for 1/2 hour, then puree with some of the soaking liquid to make a paste. I freeze it in single serving sizes, then just thaw it to serve. I often mix it with a scoop of my morning oatmeal and give it to her for breakfast, or mix with vegetables too.
There are other food additives that are naturally high in iron too, like prune juice, kelp powder, brewer's yeast, or molasses. You can add a tiny bit of any of those things to a bowl of baby food if you want to up the iron content. I often mix some brewer's yeast in with DD's food - especially if it's something like broccoli or squash - because it adds a nice savory flavor AND iron content.
ETA: There's a great article on iron supplementation and the breastfed baby on Kellymom - lots of excellent info.
Is Iron Supplementation Necessary?