Does the Book of Mormon explicitly permit polygamy?
No, I don’t recall that the Book of Mormon recommends polygamy. The polygamy among the Mormons was sanctioned by a “revelation” announced by Joseph Smith, “the American Mohammed” (as one historical account calls him). I forgot how many wives he had, but it was a fair number.
There is nothing in the US Constitution which says or implies that the states cannot regulate marriage any way they wish. It can’t be slavery, but otherwise it is up to the states.
My own opinion is that a society (such as ours) ought to be able to lay down some simple, obvious rules, or chaos will reign. One of our mores is monogamous marriage, and it is quite enough to manage.
If someone wants polygamy, he or she should move to some other country — far, far away — where this is permitted. And the destination is going to be a Muslim country, so good luck if you think that it’s going to be fun!