The Royal Institute is a scientific organization mentioned many times in the Room series. It is implied that they know about The Null to a degree, though they deny having any such knowledge.
Although the exact institution is not mentioned in the games, the Royal Institute is most likely The Royal Institution of Great Britain, est. 1799, which appears to align with the dates of the games. For more information, see this link.
Edward Lockwood had a brief employment at the Royal Institute, but the position was terminated due to unspecified misconduct.
Though she herself likely was not employed by the Institute, Abigail Lockwood mentions in one of her book entries in The Room: Old Sins that she has friends there. She attempts to reach out to them when Edward is in danger, but her cry for help is intercepted by spies from The Circle.
In The Room Three, the trilogy protagonist petitions the Institute about the disappearance of A.S., but is turned away by them for asking questions related to the Null. The protagonist suspects that they know more than they let on.
After the protagonist receives no help from the Royal Institute, they are given an anonymous tip by an individual implied to be The Craftsman. What connection The Craftsman has to the Institute may be purely superficial, but it was at their headquarters where he found and intercepted the protagonist. In a secret series of letters found around Grey Holm, The Craftsman also indicates that he had been observing the protagonist's activities even before he considered using them for his own purposes, possibly through channels at the Institute.