does not have a fixed position on consensual polygamy. Note that I mean this answer to apply to polyandry (a committed relationship of one woman with several men) as well as polygamy in the strict sense (a relationship of one man and several women).
holds that consensual relationships not based in intimidation or fraud should generally be legal, and presumably therefore polygamous relationships would be legal under the free society that Objectivism advocates.
As to whether polygamy is moral, we must consider the relevant factors.
The primary factor is individualism. Romantic love is a relationship between two individuals. To be properly experienced, it must be undertaken one-on-one. Objectivism holds that no one should accept a relationship in which they are undervalued or neglected. This has to be a danger in a polygamous relationship, since one person is presumably the sexual center, assuming the members of the relationship are all heterosexual. Traditional polygamy has usually reflected an offensive presumption of female inferiority; consider Islamic polygamy, for example, or the 19th century Mormons or Chinese. Theirs were not relationships among equals. There is also a pecking-order problem in polygamous relationships. In a pair relationship, the committed partners know who has first call on their attention. This would not be clear in a polygamous relationship, or if it were, could mean that one partner was being neglected.
Now there are broader values than romantic love that one gets from committed relationships and family. If it is beneficial to have several siblings, or several in-laws, then perhaps there is value in having more than two intimate partners in the household. Economically, for example, the household might be more robust against economic distress. This might be an argument in favor of polygamy or polyandry, or some kind of group marriage.
People may be able to find romantic love in a variety of forms, from love affairs to life-long pair relationships, to any range of polygamies or polyamouries I suppose. But it could be difficult in practice.
I think that one reason non-traditional relationship forms are still so little practiced is that they tend to be unbalanced and subject to jealousies and emotional abuse. Committed relationships that survive do so by adapting to changes in the partner's circumstances, interest, and even character over time. The difficulty of doing this must be magnified geometrically as one adds to the number of people involved, all headed in possibly divergent directions.
So I would say that Objectivism holds that polygamy is certainly wrong if it involves self-subjugation or if denies any person the values of intimate visibility that romantic relationships should provide. But there is no principle in Objectivism that rules it out in all cases. Objectivism does hold that for every person, his own happiness, in whatever rational form it takes, is the moral purpose of his life. If one's rational happiness is best served by a non-traditional relationship, considering the full, long-term context, then that relationship could be a good thing.