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You should be able to download the developer beta by going to Account -> Downloads, and then downloading the beta profile from your phone. It may be greyed out because you haven't accepted the news Terms and Conditions, which can be accepted from the main developer portal (developer.apple.com -> account). That's the only reason I could see it being greyed out.
If you go to the developer download page here, and log in with your Apple ID, scroll down until you see iOS 14 beta then either download the profile to your device or if your prefer download the IPSW file by clicking view more then downloading the file for "iPhone 8, iPhone 7"