The little permissions dialog is part and parcel of accessing the camera and microphone with Adobe Flash. You should be able to tell Adobe to remember your permission settings on that PC. To do that, right click the PoodLL audio recorder (or any flash recorder loaded from that website) and choose "settings." Choose to allow access to the camera and microphone and check the checkbox to remember those settings. I attached a screenshot to this post
Sadly Chrome now has its own set of permission requirements for camera/mic. So on Google Chrome, after you set the Adobe Flash permissions, you will see a grey horizontal dialog appear beneath the address bar/url input field. You need to also allow access from there. But it seems to remember the setting by default.