No. When the target device does not have an internet connection, The Spy App still records data but can not send data to the server.
The Spy App will save all recorded data in temporary files on the target device and wait to try to send again when the target device has an internet connection.
Therefore, recorded data will not be lost. But you have to wait until the target device has an internet connection to review new data (that is why sometimes you feel that it is slow to see the new data).