Yes, Vernier Data Share 1.1 and later allow you to download a CSV file of the data in the current session.

Tap the Download button in the top-right corner, then choose Download Data.

Products affected:
LabQuest 2.1 and later , Logger Pro 3.8.6 and later, Vernier Data Share 1.1 and later

Compatible browsers:
Chrome (Mac, PC, Android, and Chrome OS/Chromebook)
Safari 6 or later (Mac, iOS)

Note: This feature is not compatible with previous versions of Safari or with iOS 5 and earlier (including original iPad).

See also:
How do I transfer data from Vernier Data Share web app to Google Drive?
Downlaoding data from the Vernier Data Share web app gives me the error "This file type is not supported" when openign the file on a Chromebook.