The Alamy-Lightroom Bridge needs to authenticate itself with Alamy's servers if it is to do any work on your behalf. This means it needs to have your Alamy account name and password entered correctly in Lightroom's Plugin Manager.

If either of these is empty, or not correct, then of course the plugin will not be able to authenticate with Alamy.

Another possibility for not being able to authenticate are simple network problems. You can easily check by using a normal web browser and logging in to Alamy's website. Make sure that you use the same account name and password in both cases, instead of simply accepting any values that might be stored by your browser. It is easy to forget to update stored passwords!