Yes, you can download grid data like DEMs or elevation data from within Surfer. To do so:

1. Click the Home | New Map command for the grid-based map type you wish to create (eg. Home | New Map | Contour).
2. In the Open Grid dialog, click the Download button to open the Download Online Grids dialog.
   
3. Either select a layer from one of the servers in the Terrain Data category, or load your own WCS server using the instructions here. 
   
4. Choose the area to download from the Select Area to Download section.
   
5. If allowed, chose a resolution from the Select Grid Resolution to Download section.
6. Click OK to download the grid and create the grid-based map.

Note: You can select an image from a WMS or OSM server, but it will not contain relevant z information, so it is not recommended.

Note: If you select a vector layer from a WFS server, you will see this message in the Select Grid Resolution to Download section:

If you miss this message, when you click OK you will see this error:

Note: If you click File | New Map | Base, an error will be generated because you cannot use a grid file to create a base map. If you want to create a base map, select one of the other servers that provide image or vector data (WMS, OSM, or WFS types) in the Download Online Maps dialog.

Updated November 2021