|Halfway there already.|
I think this is a religion devised after the discovery of chemistry up to at least a typical modern high school level, so that the periodic nature of the elements is understood, but then explained as a pattern reflecting supernatural intervention. It's no more irrational than the interpretations of intelligent design believers who like to paint themselves as sciencey and rational. And then from that, you can branch out further into sects of the periodic pantheon that reject as much of the non-theistic interpretation as possible, sort of the opposite of the real-world progression from openly religious creationism to the rational-facaded intelligent design. My point is that it's probably believable enough that someone might develop a religion around this view of the elements.
How the periodic pantheon interacts with other religions, including presumably some older, pre-scientific ones, depends on the needs of the setting and the story, but it does seem to have one unusual implication. Most polytheistic religions in history have been quite tolerant of outside beliefs, often incorporating them into their own, mixing and matching deities like trading cards. But the periodic pantheon seems to me like it should be much more rigid, limited to only the known elements. This makes it an unusually intolerant, perhaps more dogmatic form of polytheism.
But those are all fuzzy side issues to me, things that would be determined more by the needs of the story you're trying to tell than by the essential, central needs of the pantheon concept. What I'd really rather focus on, for now, is how I see the deities of the pantheon as characters, with personalities embodying the physical traits of their respective elements. Some of this can be quite straight forward, with the iron god representing endurance and strength, generously distributed to almost everyone, while the polonium god is elusive and deadly. Hydrogen, oxygen and carbon should feature as special characters, dominant in the spheres of living things, fuels, all sorts of synthetic materials, etc. Fire is always a big deal in both religion and reality, and the idea that the oxygen god plays such a crucial role in fire would definitely earn it a lot of worship. In a setting with nuclear energy and weapons, the gods of uranium and plutonium are terrible super-monsters offering immense power, a devil's deal. Hydrogen as a Sun (and general nuclear fusion) god possibly sets up a good-vs.-evil dichotomy with the subterranean fission gods.
Beyond these individual deity characters, I also see the groups, periods and other repeating patterns of the periodic table telling neat, predictable stories about families and factions among the deities, sharing common traits. The alkali metal deities are all fiery and unstable. The noble gases don't represent nobility, but weakness, etherealness, lack of involvement. The halogens are frightening, too eager to change the world in nasty ways. The metals collectively represent similar traits of strength to iron, but not all to the same degree. The commonness of the elements in reality, roughly tied to period number, would tell us how involved in the setting the deities choose to be, or allow each other to be, or are allowed/required by some even higher law. Perhaps the older the deity, the more right it has to influence the world, while its children/underlings get to make only smaller contributions. The synthetic elements would need a fair bit of theological explaining, if your setting is aware of them. They're almost like compelling the gods to give us stuff, maybe even causing new gods to come into existence. Perhaps some sects might be horrified at this idea, while others are thrilled by it.
I'm undecided whether I prefer the interpretation that the actual atoms things are made of are literal pieces of physical gods, somehow distributed into the world, or merely the physical gifts given to the world by more ghostly, abstract gods. I think both interpretations could be embraced by different sects. In the latter case, compounds are not that interesting from a theological perspective, as that's then just crossing the line into natural science, making stuff out of stuff the gods dropped off. But in the former case, when part of the body of the sodium god touches part of the body of the chlorine god, we now have need for even more storytelling about conflict and cooperation between them as characters, and how this results in salt. You might not bother to do this with all of the millions of compounds, but a few parables and fables built around some of the most common and important compounds could flesh out the religion further. These stories give a chance to develop the character of each deity, to give them some personality and depth, or at least some dialogue for school plays to re-enact.
In a setting without the visual table representations of the elements that we're used to, this kind of semi-literal storytelling could be a feasible alternate way of explaining and remembering the periodic patterns of the elements. I wouldn't prefer that myself, I think the periodic table works brilliantly. But it's not an inevitable and unavoidable way to portray those facts, and it's interesting to think how we'd be teaching chemistry today if we'd never had a Mendeleev to come up with the table idea.
Finally, going back to the article about the naming of elements, it would probably be worthwhile to have a think about what these deities would be called, and what the elements are called if they're not sharing exactly the same names. In a setting based on our world, you'd have to either go out of your way to rename the elements, or you'd have to settle for some pretty dumb god names. Lead and Tin don't sound like very good personal names to me. Plumbum and Stannum at least sound more classical. But this is all pretty subjective. In a setting not based on our Earth, it becomes more feasible, maybe even necessary, to rename the entire list of elements. This could start drifting into conlangs. It all depends on how much world-building you want to do.