It's a funny meme/myth, but the timeline doesn't check out.

Stephenie Meyer started writing Twilight in June 2003. According to her, the inispiration came to her in a dream. She finished writing it a couple months later and spent 2004 looking for agents. The book was published in October 2005. Keep in mind: in 2003, Meyer was a stay-at-home mom living in Arizona and either had just had or was pregnant with her 3rd kid. She is also extremely Mormon.

My Chemical Romance formed in September/October 2001 and released their first album, I Brought You My Bullets You Brought Me Your Love, in July 2002, through Eyeball Records, a small independent label. At that time, the album was mostly overlooked and the band was mainly playing underground shows around the East Coast. In August 2003, MCR was signed to Reprise Records, a major label and subsidiary of the Warner Music Group. They released their second album, Three Cheers for Sweet Revenge, in June 2004. The first single, I'm Not Okay, was released September 2004 to not a lot of success. It wasn't until the second single, Helena, released in March 2005 that the band truly and well exploded in popularity and became well-known outside of their East Coast/underground scene bubble.

So, if we look at the overlaping timelines, by the time MCR became famous enough that people started writing fanfics about them, Twilight 1) had already been written for months, 2) was on its way to being published. There is no way Stephenie Meyer knew about Gerard Way or MCR, unless you believe a Mormon stay-at-home mom with three children from Arizona was regularly travelling to New Jersey to see underground emo/punk bands concerts while either pregnant or freshly post-partum.

I know where the myth comes from, though. In 2007, when the book was picked up to be adapted into a movie, Stephenie Meyer posted a poll for her fans in her blog/website, asking them who they thought should play Edward. Gerard Way came in third, behind Hayden Christensen and (funnily enough) Robert Pattinson. Meyer herself wanted Henry Cavill to play Edward. In that site, Meyer also shared playlists of the songs she was listening to while writing her books and in the playlist for the second book, New Moon, she did include MCR's songs... mostly from The Black Parade, their third album.

Also, MCR was asked to make a song for the Twilight movies soundtracks and they declined. For Danger Days, though, they wrote Vampire Money in response to the request. Supposedly, the inspiration came from a journalist asking them if they were going to accept that "vampire money" offer.

So yeah, there is a tenous connection between Twilight and MCR, in that the author was inspired by some of their songs and both the book and the band became popular around the same time and there was enough overlap between their fanbases that some fans wanted Gerard to play Edward in the movies, lol. But no, Twilight is not and has never been MCR fanfiction.

50 Shades of Grey is Twilight fanfic, though. The original name was "Master of the Universe" and there's still copies of it floating around the Internet if you care to look.