Question is what do you mean - SCP-type of organization, SCP mythology or SCP flavour?
I mean, well, I believe every one government sensible enough will have an organization, separate or some departamet, who will mess with unknown/unexplained, and essentially answer "secure, contain, protect" is the basic answer any such responsible organization will give about "what should we do with <something>". Compare SCP with "Warehouse 13" for example - and in Sixth World there is a lot of unknown/unexplained around.
SCP mythology is another matter, but also simple enough. It's quite possible to move SCP-682, SCP-073 or SCP-076. Personally I don't believe they're fit, with God, Cain and Able, but why not.
SCP flavour is the weakest point I believe. The very idea behind SCP Foundation is "unknown and unexplained usually kills, makes mad or something worse, so you better hide it, contain and protect". It's not good in a world where Dunkelzahnn elected as a President, Lofwyr is the wealthest... ahm... men on the planet, and comets creates new strains of people.
So if I wanted to put SCP into play, I'd take one of two approaches.
1. SCP WAS a thing before official Awakening. They didn't have all that things they put into Net (and they always said so!), but they have _something_, they made a great deal with damping all that "spike"-awakening things, and maybe they even kills a dragon or two and take their hoardes. But in decades after, well, SCP declined and was dissolved, their personal and riches cames to different govs and corps.
2. SCP exists, it's kind of terror group who believes that everything unexplained should be contained because it's inherintly dangerous. Look at them as Humanis Policlub, but with "unexplained", not "meta" agenda. For hillarious moments it's possible to assume that organization takes an inspiration from early-21st website. For more hillarious moments it's possible to assume that SCP from website was real thing, with website as a cover story (exactly such as site claims was).