you don’t necessarily have to download occurrence-level data to create a map. both iNaturalist and GBIF offer occurence raster tiles that you can use to make a map (https://api.inaturalist.org/v1/docs/#/Observation_Tiles and https://www.gbif.org/developer/maps, respectively). if you don’t know how to work with XYZ tiles, there are lots of tutorials available if you do an internet search, or you can look at https://forum.inaturalist.org/t/looking-for-inaturalist-observation-map-visualisation-suggestions/7322/10.
If you must download occurence-level data, and you must do it via R, you should know that rinat is relatively old and hits the deprecated iNaturalist API. it would be best if you’re downloading tons of data to get it via CSV like jwidness suggests (and then parse the CSV from R), or else you could hit the iNat API (http://api.inaturalist.org/v1/docs/#!/Observations/get_observations) or GBIF API (https://www.gbif.org/developer/occurrence) directly from R. if you’re not sure how to do that, then you could read up on that:
… or else hanly wrote a package that hits the current iNat API: