It's hard to tell by looking at your requirements what you are trying to achive. Do include a link to the API that you refer to in the future so that to narrow down the search on which API you refer to. As to the in's and out's of calling an API Chris Webb's blog contains a plethora of information on this topic https://blog.crossjoin.co.uk/?s=postcode..
One point worth mentioning is that although Power BI is feature reach when it comes to accessing various data sources, Although you can make a call to an API to get the right data, you won't be able to initiate this call from the Reporting View. So although you can get all you data you need in Power BI, unless it has already been loaded into the Data Model you won't be able to display it in any of the views. A solution that I find feasible in this case is to load some of the data in Power BI, say all of the postcodes, and then build the right URL inside your Data Model for accessing the mapped poligons so that in the Report View you can navigate to the right data via clicking through to a browser page.
