Improving upon on to what was already said: Yes they allow gay people to play much like it allows anyone at all to play, they do not however let your character to be gay; IE you have to ask the opposite gender to marry you during a quest. They do not allow you to say "gay" however, this is not for a lack of not liking gay people rather it's due to being an insult more anything much like how swear words are censored as people can be offended by them.
Just a little more insight into the matter (sorta improving further), i think that what a persons sexual or such personal preferences as what sex one marries is considered something unsuitable for runescape. Jagex does try and keep the game as kid friendly as it can without spoiling the game for everyone, and this is a subject where i don't think they care if in real life you choose to be gay but would prefer if you keep such personal info to yourself. As for having marriage included in quests, the game is meant for entertainment and any dating or marriage included in quests is just to enhance the storyline of said quests, they aren't their to start up a debate on whether or not men or women should be able to marry.
Not only that, but I think that in the time the game is set (medieval) you didn't really get gay people or homosexuality in general, or at least, they weren't thought of as gay.