And is he even there?
In Chapter 3, in the room before the frozen-over Green Room, you can go down and walk into a Zapper spotlight; if you say you're a cheater and that you're going "nowhere", then you'll be brought to a cave with a Legend of Tenna puzzle. This puzzle is connected offscreen to a puzzle after the start of the free-roam TV World area, so you must throw a box over as Susie (into the MIDDLE column, not the left) and then return to nowhere to push it onto the button. Doing so brings you to a room with a Bibliox and a console, and after doing that whole interaction, you're given a ticket which shows something "to the left of a red X". If you now enter a battle with a Ribbick, select the CroakOnX ACT, and hold left, then you'll be transported to the area with the Egg. Specifically, in this area, you have to travel up, talk to the man, say that you won't forget him (twice), and then say Yes to every other choice you're given, he'll give you the egg alongside a cryptic description of his locations in other chapters. On Thursday (Chapter 1), he "goes to sell the donuts" (probably in reference to the bake sale); on Friday (Chapter 2), he "tosses the leftovers in the dumpster" (Spamton's dumpster in the Cyber City); on Saturday (Chapter 4), he "lets it cool on the windowsill".
In Chapter 4, specifically the second dark world, there is a room with a ton of windows (the same one with the sacred moss); one of the windows has a small egg near the top instead of the normal teardrop shape, as indicated on deltarune.com/window/, which you can enter. Going in and out of this room is the transition needed to find the egg, and this can actually still be done on the Snowgrave Route unlike in Chapter 2. If you're having trouble finding the egg in the retro area, go through the middle door, talk to the man, and move down towards the lower wall (just left of the entrance) and you'll find a canvas. Interacting with this gives the Egg.
I got the egg in chapter 3. After completing it and starting chapter 4 from chapter 3's save file, it's not in my inventory in the light world...?
I have not completed chapter 4 yet.
Eggs in storage disappear between chapters. The Egg person in Chapter 3 even explains that. In Chapter 3, you should put the egg somewhere instead.