This article is from the The Religious Society of Friends FAQ, by Marc Mengel firstname.lastname@example.org with numerous contributions by others.
Officially, two Friends marry each other under the care of
the meeting, but no person "marries" them, God does. Most
meetings reserve the right to refuse to take a marriage under
their care if they feel the couple is not "clear" about their
intention to marry. Generally all present at the ceremony
sign the wedding certificate.
In the USA there have been a *few* meetings which have performed
same-gender marriages; and in one or two states for a while some
of them were even legal. This is a topic of much discussion in
many meetings, and is not something you can assume any given meeting
considers okay. Also to my knowledge the states whose marriage
laws had "Quaker loopholes" allowing Meetings to perform same
gender marriages have closed them. On the other hand, several
states are now considering allowing same-gender marriages...