A vigil beside a body would be a funeral service (actually a vigil is an all-night prayer service) held in the room where the dead body is laid out.
So, the manner in which Tom and Miss Baker strolled back into the library suggested they were going to a funeral. They were solemn, sad, and probably awkward, not meeting each other's eyes. The awkwardness is very common at funerals, and it is suggested by the fact that they had "several feet of twilight between them," and by the way Nick felt at supper a few minutes before: "wanting to look squarely at everyone, and yet to avoid all eyes."
Of course, Tom and Miss Baker were not literally about to step into a room with a dead body. This is figurative. The author, knowing it is a figure of speech, adds the words "perfectly tangible" to make it more vivid: it might be a figure of speech, but their behavior was exactly as if they were going in to view a real body. This is similar to when people add the word "literally" to heighten the intensity of a figure of speech, e.g., "I was literally dead on my feet."
The awkwardness and sadness was brought on by the fact of Tom's mistress calling the house during supper, which made it obvious to everyone present that Tom and Daisy's marriage was miserable.