Minecraft version: 1.14.4
Mod version: solcarrot-1.14.4-1.9.3, released onto curseforge Oct 21, 2019
Edit for extra detail:
Forge version: 28.1.107
Configuration file does not show up in a fresh install of minecraft through Twitch's modpacks (personal modpack). There are other config files for other mods present in "\Instances\ABA\config" (Which is the appropriate folder for this modpack), but none for SoL:CE.

I have just now also confirmed that this bug exists with no other mods present in the modpack.

Thanks for reporting this in the right place!

Can you give an example of a mod (preferably open-source) whose config is generated correctly? I'd like to compare to their code to see what I'm doing wrong, haha.

Aquaculture 2 seems to, V0.3.0
https://github.com/TeamMetallurgy/Aquaculture/
Curios as well
https://www.curseforge.com/minecraft/mc-mods/curios
Doggy Talents
https://github.com/percivalalb/DoggyTalents/
Edit: clickable URLs didn't seem to work correctly.
If any more are needed, please comment again.

That's helpful, thanks for the research! Nothing obvious popped out at me; I'll be looking into it further soon :)

I've figured out what was causing the issue: I was registering the config too late, so it wasn't generating the client one (which happens on startup). Fix will be in the next update!

Hey! Sorry it took so long; I was having some issues in the dev environment. Here is the new version on curseforge.