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).