Making Thai Khanom Tang Teak (ขนม ถัง แตก)
While I have never tried this dish (it looks delicious though!), I did some research and think I have some suggestions on veganising it.
Watching how it's cooked (in some videos I found here), it reminded me a little of 'Dutch Baby Pancakes'. These pancakes don't use a raising agent like baking powder, but rather rely on cooking at a very high heat and with a well developed batter, and are essentially inflated by steam.
So here are some tips that I might try if I were you:

Replace the egg with aquafaba. Here is a recipe for vegan Dutch Baby Pancakes that users aquafaba. About 3 tablespoons of aquafaba can be substituted per egg in your original recipe. Aquafaba works by absorbing different starches and proteins from the chickpeas, giving it egg-like qualities. It will help trap the steam pockets during the cooking process, and give your pancake the structure/strength to stay puffed up. I've found that aquafaba generally takes a little extra whipping to give it the same qualities as a whipped egg white, so maybe consider beating your batter a little longer than you normally would.
Make sure your pan is hot. Like really hot.  Try and use a pan that can become very hot, and heat evenly, like a cast iron skillet. The pan should be preheated to the point where the batter immediately starts to sizzle and set as soon as it hits the pan. Also, the videos I saw used a lid, as this will trap in the steam and help your pancakes rise.
Make sure your rest your batter. It might feel counterintuitive given it's meant to be a relatively aerated batter, but resting can help strengthen and develop glutens/proteins and give your outcome a better structure. I don't know much about rice flour (except that there is no gluten in it), but from the videos I saw, it sounds like this batter is rested for a lot longer than a Dutch Baby Pancake (which usually rests for around a half hour, maybe a bit less). The videos I saw implied a 2-3 hour rest, so I'd say try that out! I imagine it will be a process to get the rest time just right.