Sound Forge is not a portable app and AFAIK must be installed in the program files folder on the main C:\ drive. There are also necessary files in the Common and Appdata folders not to mention the registry entries. The media (audio) files can be just about anywhere.. internal, external or cloud drives. The application itself is only about 0.5 GB. You might get an 'official' answer if you email Magix, infoservice@magix.net .
btw, welcome to the Magix Sound Forge users community @EKeith-Borders.
@EKeith-Borders You can move and link every folder on your C: drive to another drive with the mklink /J Link Target command (Link and Target are common paths, for example mklink /J "C:\folder\" "D:\folder\" ). This should work fine. Just create the Target on your external drive and then run the mklink command to create the Link on C: and then install Sound Forge inside the Link.
. . . . I have an external ssd with 512 gb of storage hooked up via usb. . . . .
Technically, the program can be installed to a second internal hard drive.
Installing to a USB device is not recommended. USB ports are shared devices, they can also be shutdown by Windows power management, you can override this, however the drives themselves have a power saving mode which you cannot override.
In addition there are still parts of the program installed onto the C: drive, while you can make the changes @SP. suggested these can cause further issues with Windows Updates and possibly the program updates as well.