The idea that role playing involved a property called "immersion" occurred to the early adopters of the 1970s fairly early. The earliest explicit use I've found was that of Pieter Roos, as shown in this excerpt above from The Wild Hunt #15 (1977), where he identifies it as the overall goal of playing RPGs. Before the end of the decade, calls to "immerse yourself" began to appear in games and modules published by TSR. The Elusive Shift pays particular attention to the uptake of "immersion" as a term because of its relationship to how people in the 1970s understood the nature of role playing.
Almost as soon as people started experimenting with Dungeons & Dragons, they recognized the potential for losing yourself in a role, feeling in some sense as if you were actually in the situation of the game character. Sandy Eisen, a D&D player at Cambridge University in 1975, reported that as a beginning player, he felt like he was really "living the part" and that through "willing suspension of disbelief" he found himself "in the dungeon."
Eisen did not have any particular word for this property (no one was even saying "role playing" then), but he found it compelling enough that he vowed that when he ran D&D for new players, he would not tell them the rules -- he found that understanding the system bogged him down in "wargame mechanics," rather than focusing on the "real-life considerations" that a person in the game situation might. This does have some precedents in Kriegsspiel, in how wargamers would relay troop orders to a referee in order to experience the closest approximation to actual command, but Eisen valued it for a unique way it made him feel... even if he didn't know quite what to call it.
Two years later, Kevin Slimak wrote in The Wild Hunt #12 about role playing a character such that you "can put yourself into his place, submerging yourself to the extent which you are able." He referred in particular to acting within the rolled characteristics of characters -- like not making smart decisions for a stupid character and vice versa -- which is at least requires some rough understanding of the system. This idea of "submerging" your personality into that of a character also has some antecedents in wargaming, especially in the lead-up to the release of D&D. Mike Carr, designer of the First World War aerial wargame Fight in the Skies, had recommended that a player direct his pilot to "perform according to his personality, not yours," and recommends developing "Personality Profiles" and backstories for pilots that would help determine how their characters should be played.
When we begin to see appeals to immersion in TSR publications, maybe we shouldn't be surprised to find them coming from Carr. The tips for players in his module In Search of the Unknown (B1), for example, concludes with the following:
Similar language can be found in Boot Hill (1979), which Carr edited. Carr clearly connects "playing your character's role" with his call to "immerse yourself in the game setting," but no one really explained exactly what that immersion entailed. Maybe it meant something like what Ed Greenwood talks about in Dragon #49, speaking very much to the properties that Eisen sought back in 1975. Greenwood explicitly positioned "role play" as the answer to the question "How can one play a game without knowing the rules?"
"As a player, state what you (the character) are trying to do, and the referee (who knows the rules) will tell you what is actually happening," Greenwood explains. As late as 1981, Greenwood still knew well the roots of this practice in free Kriegsspiel, and was aware that that it had attracted controversy, as referee decisions could seem "arbitrary" to players. And indeed, this approach to role playing has always had its critics. While Greenwood does not directly label his approach "immersion," that is perhaps for the best, as we have been told to "stop saying immersion" since it can mean so many different things: whether Roos, and Eisen, and Slimak, and Carr were talking about precisely the same thing is debatable. But their attempts to "live the part," to "take on your character's persona," played a significant part in how the practice of role playing defined itself.