Hi E.,
First off, I just want to offer you some support on your dairy- and egg-free journey. My husband has a dairy sensitivity, and I think our son might, too. He is on the cusp of being ready to get whole milk (9 months). We tried it once, and it did not sit well with him.
There are lots of options out there, but you're not going to necessarily see them at Cub and Rainbow. For example, there is soy-based yogurt and cheese, so your daughter could continue to eat her favorites.
I see you are north of the Twin Cities, but I would recommend that you go the The Wedge Co-op in Minneapolis. If you were able to go on a weekday morning or afternoon, you won't have to deal with traffic, there won't be very many other shoppers, and the staff will be able to anwer your questions. Everyone who works there is quite knowledgable and happy to help. There are other co-ops in the TC area, but I think the Wedge is the best in terms of Customer Service, and I make a special trip out there once/month.
With the awareness of allergies and sensitivities, coupled with the vegan diet (people who don't eat ANY animal products as a lifestyle choice), there is so much on the market now. The Wedge even has an extensive vegan bake case, so your little girl can still have her dairy- and egg-free cake, and eat it, too. ;-)
You've received some good advice on supplements, and I also think that is something you should look into. If you are overwhelmed by what's out there for information, a one-time visit to a Naturopathy Doctor might be money well spent. My family has seen Dr. Amy Johnson-Grass before, and we loved her. She could help you set up a diet and supplement plan for your little girl. Unfortunately in MN, visits to an ND are not covered by medical insurance, but you *can* pay for the visit with HSA or Flex Plan dollars, if you have either. I felt her fees were reasonable ($150), considering she met with us for a full 90 minutes and we walked away with more information that I ever could have found on my own.
Good luck to you!