Why can't the transaction log shrink during a backup?
The transaction log is used during the backup process to store all the changes that were committed while the backup was being taken.  These change are applied to the backup file so that the backup is consistent as of the time it finished (only committed transactions).  

But three days for a full?  I backup a Terabyte database every day and it only takes 90 minutes (compression, multiple files, SQL native).  Given your version, you should look into backing up to multiple files at the same time (or speeding up your backup storage).

Answer: the transaction log is in use by the backup process.

Because you were sending this to the vendor, I suspect that you sent it directly from your local SQL Server to the Vendor on a fileshare (especially since you say it only takes 90 minutes to backup locally).  

For this scenario, I would recommend that you backup locally first (and if you were in FULL/BULK recovery mode, maybe with the COPY_ONLY flag) and then send it to the vendor.  You won't tie up your SQL Server with this backup and you gain the opportunity to compress the backup using 7zip or similar (which should easily cut it down by 50%).