I hope you've read the closely related answer (here).
Intensive sport training, and therefore weightlifting, demands higher levels of nutrient intake. The nutrient that is most likely to be insufficient in people doing such kinds of training is - as you pointed out - iron. The first thing an athlete has to do is to periodically check the blood status, and it seems you already did it. Since your doctor said you're lacking iron, you should consider increasing intake. You should consider:
- consuming more iron-rich vegetables (such as lentils, beans and green leaves),
- consuming more vitamin C-rich fruits and veggies (such as oranges, lemons, etc.). Vitamin C increases the absorption rate of non-heme iron, the type that is found in vegetable sources.
- consuming less, or avoiding, foods that decrease your iron absorption, such as coffee and alcohol,
- reducing or quitting dairy, since calcium competes with iron carriers, in other words excess of calcium might increase the iron deficiency,
- take care of your gut, since a healthy gut is the best way to optimize nutrients absorption; and a gut that is not healthy might lead to deficiencies even without sport training. You take care of your gut by eating a good amount of vegetables and fruits, that help you with fiber and vitamins.
Additionally, some sources refer to gluten as a possible concurrent cause of nutrient deprivation, although the matter is still highly disputed. Gluten might alter gut permeability1, and hence alter nutrient absorption rates2. However the problem is still under heavy study and has no final answer yet.
Finally don't forget to test also the blood status of other key nutrients, such as vitamin B12; in this case you should test vitamin B12 blood status, folates and homocysteine levels.
If after the suggestions that I gave you, you still have iron deficiency, you might consider using an iron supplement, but you should seek the advice of a nutritionist. Considering that many professionals have an old-school background on vegan/vegetarian diets, and might have some prejudices against it, I would recommend you speak to a health professional who has a open mind about that.