How do I determine whether a drive is an HDD or SSD in Windows Vista or 7 without 3rd party apps?

There actually is an easy way to do that from Windows 8 and up:

1. Right-click on the partition you want to check. Make sure it belongs to the correct disk.
2. Check Properties
3. Select the second tab, Tools
4. Choose Optimize

The new window should list all partitions and the type of the corresponding data container (here Solid State Drive and Hard Disk Drive).

I would probably have the user run wmic from the start menu then have them type diskdrive get caption

wmic:root\cli>diskdrive get caption
Caption
HGST HDN724030ALE640
Samsung SSD 850 EVO 500GB
Kingston DataTraveler 2.0 USB Device

"Do any of them say SSD?"

If it's not obvious, then do the googling you mentioned. This is the least intrusive solution to the user, and the easiest to communicate, in my opinion. And since this is WMI based you could do it remotely in a domain environment.

This only works on Win 8+ https://technet.microsoft.com/en-us/library/hh848686%28v=wps.620%29.aspx

get the user to somehow open PowerShell, and let him run the following Command.

example output:

PS C:\Users\xy> (get-physicaldisk).MediaType
SSD