Is it possible to post data retrieved via a Web API into a SharePoint list? my scenario is that I want to pull some data from a third product via its WEB API and post it if possible into a SharePoint Online calendar. As I've not done this before, I'm not sure what my options, if any, are.
It is definitely possible. Your question does not indicate what approach you are using to develop your program so I will keep it more general. You should take a look at the SharePoint REST service info here Get to know Sharepoint REST Service
Other than that, Danny is correct that authentication is the other issue to consider.
SharePoint has a REST API just like any other
The main issue is authentication,
so it depends on whether you are already logged in
If not then you need to authorize with O365 and push your content
If you are logged in you only need to read your Web API and use the SharePoint REST API to write the data.. search for REST and SharePoint