As someone who rather likes the totally non-canonical idea of gay Edmund, there is really no textual evidence to support this idea and you are right to point out that it is extremely unlikely that Lewis intended the character to be gay. There's not even much of what most people would consider obvious gay subtext. We don't see Edmund longingly describe the appearance or other attributes of other male characters, for example. We don't see him form the kind of extremely close male friendship that seems to border on romantic that you sometimes see in other works. We don't really see other characters describe him in ways that are coded gay.

All that being said, the person in the linked article isn't the first to come up with the idea of gay Edmund. A quick look at Narnia's M/M category over on the popular fan fiction site, An Archive Of Our Own reveals that Edmund is the character most listed in male/male pairings section despite the fact that both Peter and Caspian are played by older, arguably more attractive actors in the movies. Caspian/Edmund is the most popular male/male romantic pairing.

So why do many fans see Edmund, in particular, as gay but not, say, Peter or Lucy? What you have to realize about the way that many LGBT people read or view fiction is that they sometimes connect aspects of a particular character's journey or arc to their own journey or struggles with queerness.

One rather prominent example of this is the character Elsa from the Disney movie Frozen. A lot of LGBT people read Elsa as gay, not because she pines after women, explicitly or otherwise, but because her character arc shares much in common with the type of struggles that LGBT people face. Elsa is forced by her parents to keep her powers hidden, "conceal, don't feel." This causes her to feel alienated from her community and family, even from her sister who loves her no matter what. Elsa eventually chooses to live alone in a literal ice fortress because this is the only place where she can be herself. This parallels the issues that many young gay people face with lack of family acceptance, feelings of alienation and isolation, feeling somehow dangerous to others, suppression of a major part of themselves, and choosing to be apart from the community that they were born into because they don't feel they can be themselves there. But someone could, of course, correctly point out that there is nothing in the movie itself that suggests Elsa is gay and her sexuality goes entirely unaddressed in the movie.

And I do think there are elements of Edmund's character that LGBT people might identify with in this way. Edmund starts out the series feeling alienated from his siblings for reasons that aren't entirely clear. Edmund started being a jerk the first time he came back from boarding school which perhaps indicates that he was experiencing some form of bullying (interestingly, the same boarding school doesn't seem to have had a similar effect on Peter -- and though I believe Peter suggests that Edmund himself has become a bully, it wouldn't be unusual for someone who is bullied to, in turn, bully others). Edmund seems to harbor a certain amount of resentment toward his more explicitly masculine older brother (this isn't to say that Edmund is feminine, just that Peter is usually given the more explicitly masculine tasks of physically protecting the girls or generally waving his sword around, for example). In his resentment, Edmund screws up big time and betrays his siblings. Edmund is forgiven, but aspects of this forgiveness are kept "secret" -- the books explicitly point out that the all the conversation between Edmund and Aslan was never revealed to others, but that it was a conversation that Edmund never forgot. Edmund ends up doing something heroic in a way that suits his personality (using his smarts to come up with the idea of breaking the witch's wand -- something that no one else thought of).

I'm not saying that all gay people identify with these experiences or that no straight people can identify with them, but I think there is a certain overlap between Edmund's experiences and the experiences of many LGBT people that isn't really true of the other Pevensie siblings. The feelings of alienation and isolation followed by eventual acceptance and affirmation of Edmund's particular qualities as a person is a journey that would feel familiar and comforting to many LGBT people. Actually, now that I think of it, Edmund and Elsa share quite a bit in common -- the feelings of alienation from family, the somewhat closed-off personality, the much more open, optimistic, sweet younger sister, even the association with cold and winter (Edmund goes to the White Witch's castle to ally with her, Elsa has ice powers and goes off on her own to create her own ice castle).

And to answer this question: Conversely, is there a way to interpret the text that suggests he's not gay? No. There is nothing in the text itself that suggests he is heterosexual. We see more of adult Edmund in The Horse and His Boy than we do the other Pevensie siblings, but we don't see anything that would point to a possible sexuality. This certainly doesn't mean that he's gay -- but it is one less obstacle to people viewing him in that way.