The Notes API is described on the main API page API v0.6#Map Notes API
It is OK for third party sites or apps that use OpenStreetMap data to include notes functionality through the API. This feature will however, only be useful if the quality of reports are high. Therefore it is important that the issue reports include sufficient information and detail for an experienced mapper to be able to fix the issue. Also, it is important that you make your users aware that this is to be used only for commenting on map data issues and not general aspects of your site or app.
Furthermore, it is important to remember, that this functionality is intended for humans to communicate with other humans (mappers). It is not a place to dump automated error checking.
You may recognise "notes" as a similar concept to OpenStreetBugs, which ran as a separate website and database. Integration of OpenStreetBugs into the main website had been discussed ever since OpenStreetBugs first appeared. The launch of "notes" brings this integration along with enhanced email contact features. OpenStreetBugs has now been phased out - congratulations to all involved in the transition!
Being part of the website, the code and issues list can be found under OpenStreetMap website on GitHub.