while this is possible within iNat and affiliate pages, i’m fairly certain that iNat disallows cross origin use of most of their webpages. so you can try, but it probably won’t work inside an iframe in a third party site.
you probably need to create your own map using iNat’s API and your own basemap service. you can look at the GitHub repo referenced in my profile for examples of such custom maps. you’re welcome to adapt the code there, but don’t embed my pages directly in your page. (if i find that folks are embedding my pages in their own pages, that will force me to add some framekilling code in them, which i prefer not to do.) you can also search the forum for “mappiness”, and that will take you to a thread that explains how to make your own custom maps using the API, which may be helpful if you’ve never worked with that sort of thing before.