Yes, with a backup file. TaciVox keeps its data out of iCloud and iTunes backups on purpose, so a normal iPhone backup doesn't carry it over.
On the old iPhone, go to Settings › Back up everything. TaciVox makes one file with every transcript and recording, locked with a passphrase it shows you once. Write the passphrase down.
Save the file to Files, a computer or a USB stick, or AirDrop it to the new iPhone. Then set up TaciVox on the new iPhone and go to Settings › Restore from backup.