It has been noted that Muslim women who wear garments that cover from head to toe may suffer from Vitamin D deficiency and obesity. Now, suppose a Muslim girl and her family moves to a Western country. Is it permissible then in Islam for a girl to leave her burqa, or must she wear a burqa at all times as an observant adherent? Is there any wiggle room in the law? Can the girl wear a hijab instead so that her face is shown in the sun?
First of all I want to say that the burqa is a cultural dress, not the Islamic hijab. I am not saying that the woman who wears it should not wear it as it is the best dress for practicing hijab.
Secondly women wears burqa only when they are out of home, when in front of na-mahram/non-mahram and not all the day. Therefore I found all those research as ridiculous and fishy, they are just a propaganda against Islamic dressing.
So even a person of just average common sense can make out that there is no relation between burqa and vitamin deficiency/obesity.
And based on his answer a woman can show her face in the West.
Women in Arab; women in West
You're taking this to the wrong side a bit. Islam was never a religion for just Arab people. Islam is a complete code of conduct for humanity. So what do you expect? A woman from Arab will have to wear the Hijab where its alot hot and humid temp, and a woman won't need to wear hers burkha in West where temp is bearable just because her complexity is being not fair? Just because her doctor is telling her that remove the burkha and you'll be fair in color? How can that happen. Allah would never do injustice to his creature.
What is burkha
Burkha was never meant to be something that is black in color; which starts covering female body from her head, lets say hairs, and goes down all the way to her feet even covering her shoes. That's a cultural burkha not the islamic one. That burkha was used by people of tribes. They tried to hide their women from anyone, their females had to wear burkha even in the homes.
Where to wear a burkha
Islam wants a woman to be covered when she is infront of a person that not a mehram for her. Lets say, female's father, brother, mother and all these relatives don't need to meet her only when she is wearing her burkha. Burkha is asked to wear once she is outside where anyone from public can see her. Then burkha needs to be wore.
Living in West? still wear burkha
Going to west doesn't mean to leave your burkha; burkha is a part of our Islamic culture, not the part of eastern culture! :) However, even if a girl is wearing Hijab she is half covered; that's enough for her if she feels uncomfortable in burkha. You need to take your culture with you, and keep it in your heart. Respect your culture like they do respect their own.
What I might share
The best answer would be, Allah knows how his creatures feels in a condition. And will never punish them for the sins or discrimination of the religion that they have no control over. Just like you mentioned Vitamin D and all other symptons, I am not well aware of these things but if a female feels uncomfortable, she can replace burkha with Hijab, but she must be covered.
First of all, hijab is the actual type of covering, which woman should do. It has some basic criteria that except hands and eyes, whole body should be covered. (Most scholars recommend covering the face as well, although that does not make it obligatory, but iff done it's mutabahab). And there is no "burqa" in Islam, although the burqa is one form of hijab.
In any case, a woman is NOT allowed to leave hijab, UNLESS she has to face very serious consequences for it or following it may create serious issues for her. Otherwise, leaving hijab is in no way allowed in Islam.
Also, you have to know that you don't have to do complete hijab in your home. In the boundary of home, you still have a lot of sunlight coming inside your home, so can't you get sunlight from there?