How to convert from a webmap on ArcGIS to downloading shapefile format?

Are you the owner of the web map and data layers in the map?

I haven't ever seen an option to download a whole web map that actually works without the owner packaging the map. Usually, you'd open ArcGIS Pro and login to ArcGIS Online through there. If the web map you're trying to view locally is shared you can find it by opening the Catalog pane, clicking Portal, and searching for the web map's name.

Another option if you are the owner of ArcGIS Online content: back in AGOL web map, you can click the layer's "more options" ellipsis -> show item details. This opens up a new window in your web browser displaying the layer's content page. There will be an option to "Export Data" on the right where you can choose to download in shp/KMZ/CSV formats.

If you aren't the owner, then contact the owner and ask if they can enable others to export data. This is a checkbox option under the layer's settings.