There is no one body that is accepted by Muslims anywhere (and this could be more of a blessing than a hindrance). There has been a trend over the past half-century or so to form councils that issue fatawa in response to issues that come up. These councils accept pertinent questions, discuss it among themselves, and finally issue their ruling. Some are well-accepted across the Muslim world and some limited to their geographic region. For the most part, they are by country. Some examples include:
Islamic Fiqh Council of the Muslim World League
Dar al-Ifta al-Masriyyah
Ifta at any Darul Uloom in India, Pakistan, South Africa, England, etc.
Fiqh Council of North America
Assembly of Muslim Jurists of America
European Council for Fatwa and Research
Lajnah Da'imah (Saudi Arabia)
and many more. The fiqh councils of Azhar, Arabia, and Darul Uloom have large followings all over the world. The others are mainly restricted to their respective regions.