There is no DOI associated with using the API. You’ll need to publish some sort of table/appendix or other way of citing the observations individually that meets the CC license guidelines for each of the observations that you use.
There are some other posts that may be of use:
https://forum.inaturalist.org/t/data-usage-is-it-ok-to-include-observations-on-a-website/21173
https://forum.inaturalist.org/t/citing-inaturalist-data/8684
https://forum.inaturalist.org/t/questions-re-use-of-inaturalist-data-in-research-gbif-exports-licenses-of-observations/21400