I have been a vegetarian since birth. I do not drink coffee, alcohol and I also do not smoke, so I am very much like you in that regard. I normally do not experience cold hands, but sometimes cold feet. In fact most of my family members and co-workers always ask me why my hands are so hot. Sometimes when I Water Fast I do feel cold all over. Usually cold hands means low iron.
You take supplements, are they conventional from Vitamin stores? Most of those are synthetic.
Kudos to you that you are a vegetarian and staying away from all the junk foods. Try apple for breakfast an observe if that warms you up. Also raw garlic, two cloves in the morning and at night shall help to warm your body.
Flax and Walnuts should be actually very helpful to you. They are not the contributors to your cold hands.
Hatha Yoga with hand movements will also be helpful.