Suppose I discover an uncatalogued crater on the lunar surface, and decide to name it. How would I submit my proposed name to the IAU?
You use this from https://planetarynames.wr.usgs.gov/FeatureNameRequest. You need to follow the rules as set out in https://planetarynames.wr.usgs.gov/Page/Rules as well as following the themes for naming Moon based features https://planetarynames.wr.usgs.gov/Page/Categories.
To directly answer your question, the other answer is correct: The United States Geologic Survey's Astrobiology office handles ad hoc US submissions to the International Astronomical Union's nomenclature committee, so their form is what you would need to fill out. Nomenclature across the solar system has been set by the IAU to follow themes based on feature types and sizes, which must be followed, and one also needs to follow the IAU rules and conventions on assigning names.
With that in mind, planetary nomenclature is not something that someone just opts to do. Just because you see something on a planetary surface, that does not mean (a) you can name it, nor does it (b) mean that it even will receive a name. As the third link notes:
The number of names chosen for each body should be kept to a minimum. Features should be named only when they have special scientific interest, and when the naming of such features is useful to the scientific and cartographic communities at large.
So if the feature you are interested in naming does not meet those criteria (in addition to all the others), it will not receive an official name. The purpose of this sort of restriction is to keep things simple. Because of this, names really are only considered when submitted by members of the scientific community, since they are the ones who most commonly use them. That is why the request form (first link) requires you to put your professional affiliation and why students must have an accompanying letter of support by their advisor. Similarly, that form requires a statement of justification that explains why the feature to be named, and the name itself, all fit those requirements.
With respect to craters, are you sure that what you have identified on the lunar surface is uncatalogued? Speaking as someone who does crater database generation as part of my day job, if the crater is larger than about 0.5 km, it's probably already been catalogued by me or someone else, and there are numerous not-so-obviously-published databases that include hundreds of thousands of smaller craters. That also means that caution should be used when saying that you have found something not yet catalogued by someone else. There was an AI-crater-detection paper that got a lot of press about two years ago with headlines proclaiming, "AI Discovers 6000 New Craters on the Moon" or variants thereof. The problem with that headline is that those craters simply were not in an incomplete database they used for comparison, but they were well catalogued by numerous other people.
As a side note, while there are >2 million craters in my public database, it's important to note that the IAU has only formally named 1628 craters, at the time of this posting, due to those restrictions of "useful to the scientific and cartographic communities at large." Even the newly discovered craters seen by the Lunar Reconnaissance Orbiter, which often get press releases and are of interest scientifically, are not named.
Finally, my answer is not meant to discourage you, but rather to explain the process and the reason behind it, and to caution you when you say that you have discovered something not previously catalogued – proving a negative (that it's not been seen before) is quite difficult.