Hi Catie,
Are you working in ArcGIS Online? Is your list a CSV file or can you save it as a CSV?
If both of the above are true, I think the easiest way to accomplish this would be the following workflow:
1) Add a general Zip Code layer (https://www.arcgis.com/home/webmap/viewer.html > sign in > Add > Search for Layers > USA ZIP Code Areas > add this to the map.
2) Then Add > Add Layer from File > your CSV/Excel File > match your zip code field to the Postal location type > Add. This will generate a point layer of zip code centroids.
3) Join Features in the ArcGIS Online Map Viewer. Make sure the polygons are the Target Features and the points are the Join features. Choose Intersects as a spatial relationship, then accept the rest of the defaults. Make sure to check credit usage beforehand by clicking on "Show credits."
4) Symbolize the output layer differently on top of the generic Zip Code layer.
Hope this helps,
-Peter