Thanks to a post on Reddit, we've learned that Epic Games currently have no plans to put their store on Linux.
In response to a user question about it on Twitter, Sergey Galyonkin, the person behind Steam Spy who now works for Epic Games as the "Director of Publishing Strategy" said this:
It really isn’t on the roadmap right now. Doesn’t mean this won’t change in the future, it’s just we have so many features to implement.
It's interesting, since their original announcement mentioned the store was coming to "other open platforms" besides Windows, Mac and Android which we presumed would mean Linux. It's odd, since there aren't really other open platforms besides those to put a store on. We also had Tim Sweeney, the founder of Epic Games, give a ray of hope on Twitter with "We'll See :D" in reply to a user asking if the other open platforms meant Linux. So, I do still find it very odd that it's not on the roadmap at all. Not surprising though, Linux has always been low priority for Epic Games.
This could create an issue for us in future, since Epic Games are taking on timed-exclusive games which would mean no possibility for a Linux version until that ends. Even then, the developers of those games could decide to remain solely on the Epic Store. Remember, this has already happened with Satisfactory from Coffee Stain having the Steam store page removed to be exclusive to the Epic Store.
Speaking on Reddit, Epic's Sweeney said "These exclusives don’t come to stores for free; they’re a result of some combination of marketing commitments, development funding, or revenue guarantees.". So with that in mind, Epic Games are offering some commitments to developers to get their games, which could sway some future high-profile titles away from the likes of Steam.
For Linux gamers we still have Steam, GOG, Humble Store, itch.io and a few others which support Linux games. Heck, even Discord confirmed their store will come to Linux. For now, the Epic Store isn't a huge deal and doesn't have a lot of games (or features) for users so it won't be a huge problem right away. It will be interesting to follow, especially to see what Valve plan to do to prevent too many games leaving.