Basically, as far as I can discover, no, though this hasn't stopped numbers of people going through marriage ceremonies of no legal standing with animals. A teenager in Bali was forced to marry a cow with which he had had intimate relations. He claimed the cow had flirted with him. In 2009 in Ghana a woman married her dog because in character he resembled her father. We must not forget the wedding in 2006 between Sharon Tendler and a dolphin, which took place in Israel. Barbarella Buchner of the Canary Islands has been married to her two cats for ten years. The publicized marriage between a man and a goat in Juba, South Sudan, in 2006 was, I suspect, not a serious event.
I have also read of an Anglican vicar who married a woman to a tree. I have not asked the Church of England for a statement on its position on dendrogamy. Years ago, an Australian minister married a man to his television set.
There is even a website called marryyourpet.com This is presided over by a lady called Dominique, who managed to get herself ordained in some religious denomination and has married hundreds of persons.
I would point out that, as far as I know, none of these marriages has legal status.
If you don't want to marry an animal, you could always try sologamy. In this, you marry yourself . The first person in the United States to do it seems to have been Linda Baker in 1993. She had seven bridesmaids and her wedding was attended by 75 guests. This practice too has a website IMarriedMe.com Insofar as I am aware, these marriages enjoy no legal status - yet. Sologamy seems to be a definite trend. I would warn anyone entering into such an arrangement that subsequent divorce might cause problems.