Goosechase does require players to have an internet connection during the game. This is because submissions are uploaded right after they are taken so that everyone can stay up to date via the Activity Feed and Leaderboard.
As an alternative, you can take and save pictures/videos to your device's camera roll and upload them once an internet connection is available.
Please note that not all game Creators allow for camera roll uploads for all Missions. Please check with the organizing group of your Goosechase to see if this is an option.