Navigate to 'System Preferences > Security & Privacy > General'
Click the lock on the bottom left of the window and enter an admin password
Click the radio button that says 'Anywhere'
Click the lock in the bottom left of the window again to prevent further changes
You would have to remember to go back and change it though when you're done downloading applications from that specific website/developer. Yes, it's a pain, but you could set a reminder for yourself with a date on it so that your Mac can remind you to change it back at a certain time!