You do NOT need to have bowel issues to have celiac disease, some people have it their whole life and never have symptoms until into their 70’s!
I am surprised that your doctors have not run a celiac panel on you in attempts to figure out why you are anemic. The screening is a simple blood test, I would certainly ask for one.
Personally, one of my first symptoms was anemia, rls, and migraines. I did not develop gastric symptoms until about 10 years into the disease. 5 years later I developed severe lactose intolerance, you don’t want to wait for that to start happening and some people never become lactose intol. anyway.
Go here for info on the celiac screening bloodwork. Malabsorption can certainly be a reason for all sorts of vitamin deficiencies. Has your doc done any other bloodwork for Vit A, the B’s, or D def.? If those are low in spite of lots of time in the sun and eating lots of rich foods, time to get the panel asap! Remember, you can all the symptoms or absolutely none.Go here:

Coeliac disease also called Celiac Disease involves the epithelial (lining) of the small intestines, which is caused by an autoimmune response to the intake of gluten containing problems.

Do you encounter and have you be been diagnosed with bowel problems? If yes, than maybe.

You mentioned Iron deficiency anemia, is this what you have been diagnosed with? As you may know, there are several factors which cause low iron levels in the body, including an immune response against a pathogenic organism.

Unfortunately I can not make any recommendations for you; you can utilize your acquired knowledge and gently suggest possibilities to your doctor.