I am frequently asked about whether or not Power BI supports exporting of data. The short answer to this question, is “yes, Power BI does support exporting data from visuals.” There is a good article on this topic here: https://docs.microsoft.com/en-us/power-bi/visuals/power-bi-visualization-export-data.
However, there are some “limitations and considerations” to this export feature that tend to impact us in the public sector more than organizations in the private sector. In fact, the limitation I encounter most often isn’t even listed in the article I referenced above; exporting of data from a visual is not supported in reports that have been shared using “Publish to Web”. For what it is worth; this limitation is explicitly called out on the “Publish to Web” documentation page: https://docs.microsoft.com/en-us/power-bi/collaborate-share/service-publish-to-web#limitations (although I think it should be documented on both sites).

This means that if you are using the “Publish to Web” feature in Power BI to share reports with your “public” users, they will not be able to export the underlying data. If you would like to be able to allow your users to export the underlying data, there are a couple of options.
First, if the report is hosted in a Power BI Premium capacity (or if the user has a Power BI Pro license), users could be able to export the underlying data if you wanted to allow this. Note: the export from visuals features does not allow for “unlimited” export of data; there are limits to consider: https://docs.microsoft.com/en-us/power-bi/visuals/power-bi-visualization-export-data#limitations-and-considerations.
Another option is to host the data outside of Power BI (e.g., Azure Blog Storage) and then leverage hyperlinks in the reporting to allow downloads (i.e. exporting) from the data repository.
I generally recommend the second option as it gives you much more control over the data “export” process and is usually less expensive than Power BI Premium licensing.
Regardless, it is important to understand that if you are using Publish to Web to share reports with your external users (e.g., citizens, partners, media, etc.), those users will not have access to the underlying data. If you also need to share the underlying data, you have a couple of simple options for including the data with the reports you are sharing.