Officially, the curator guide states:
Common names for animals should be entered in title case with each word starting with a capital letter (e.g. Atlantic Bluefin Tuna) but not the second part of adjectival hyphenated words (e.g. Sulphur-crested Cockatoo).
Common names for plants, fungi, and other non-animal taxa should be entered in lowercase (e.g. spotted gum) except for proper nouns, which should start with a capital letter wherever they appear (e.g. Tasmanian blue gum).
NOTE: For reasons of style, iNaturalist often automatically displays common names in title case. However, common names should never be entered in sentence case (e.g. it would be entered as “mountain grey gum” not “Mountain grey gum”).
People don’t worry about this style anymore because iNaturalist now officially capitalizes vernaculars as entered (as cmcheatle stated above). However, a discrepancy remains where names are still displayed as entered on the mobile app, and until this issue is fixed, I still enter non-animal taxa names as lower-cased for consistency.