No.
Direct access to the NAS is not possible. The software must be running on the PC and the NAS drive must be connected to the PC.

It is possible to save directly to the cloud (e.g. Onedrive, Dropbox, Google Drive). Many NAS support a sync with a cloud. Theoretically, it would be possible to access the NAS without a PC.

It is now possible with the new version 2.12 of the ScanSnap software:

"A new function has been added so that scanned images from the ScanSnap iX1600 can be saved directly to a network folder without using a computer."

(4 years later, yes)

My ScanSnap iX1600 (purchased Dec. 2024) can scan directly to the NAS share.