Apparently it depends on the app or input field. The generated emoji is an image, not a unicode string, so the input must be able to accept inline images.
You can send them to yourself in Signal, for example, then download the image and post it here.
I'll be the sacrificial dumb guy here. Are the posts here just copy/paste, or is there a way to force some emojis together? Secondary question, how do I just type in a Unicode value into my phone keyboard to get an emoji?
Ok. It took some research, but I think I have it worked out. You have to either be using the gboard keyboard or go to the emoji kitchen website. From the website, you can just combine, copy, and paste. From Gboard, you need to select the smiley face next to the space bar OR click the 4 squares and then go into "emoji." From there, you can select 2 emojis (assuming they are compatible) to create a new emoji, or select 1 and choose one of the recommendations. I can't get it to work from my phone in the Voyager app, but I also can't guarantee that's Voyager's problem vs. mine. It does work on my texting app, however, so I know it works. For further reference.
This reminds me of those games where you start off with water, wind, earth and fire and combine them to make new elements, which you then combine to make new elements, etc.