There are only 51 observations of Mantophasmatodea, and by only 28 observers, mostly in South Africa. So (unless the observer lives in South Africa) that would be one of the harder orders to find.
EDIT: Siphoniulida, a monogeneric Order of millipedes, have exactly zero observations on iNat. It contains only two known extant species, one of which is known from only a single specimen, and the other from only a few. So I’m going to say this would likely be the hardest extant Arthropod order to find, and would make your quest potentially take a very long time.
Wikipedia lists about 138 extant arthropod orders. With 5 minutes per observation (for taking fotos and upload) one needs about 11.5 hours to upload 138 observations.
this is certainly a drastic simplification, as finding arthropods of all 138 orders is a in fact a whole other issue, Did you check whether or not there is somebody on iNat who already managed to observe more than 100 orders?
filter the dynamic life list by arthropods, export the list, and then in your own tools filter the exported list by taxon rank = order (or summarize the results by rank).
if you need to do more complicated initial filtering, you can use https://jumear.github.io/stirfry/iNat_observations_taxonomy to get results similar to those from the dynamic life list, with more options for filtering observations (in the URL).
that’s not information that you can get from the system in one query. you’d have to figure it out by getting a target list of potential candidates and then getting that information from each of their life lists.
if it’s just to satisfy your personal curiosity, you’ll just have to decide whether you’re curious enough to do the work to figure out the answer.
But this still only includes observations that are only identified to order level, not all observations of stuff in said order. And clicking the observers tab still gives me how many species each person observed, not how many orders
like i said above, “which user has observed the most arthropod orders” is not something you can get in one query. it’s going to take a lot more work to get the answer to this.
if you can’t figure out how to get a count of orders for any given person, i also described how to do that above.
I made an assumption that the people who have observed the most arthropod species have probably also observed the most orders, here is a table from the top 10
Going through 10 peoples life list, opening all arthropod dropdowns to order and then crtl+f searching for “Order” and making this table took about 15 minutes.
If you truly want to try to find the person with the most orders observed, I suspect they are in the top 100 users of arthropod species observed.