No, you don't have to be registered in the Hall of Fame to catch the three Regis. The only conditions are;
>The Regi trio have got the same legwork to activate. Before the three locations become available to you you must have a Relicanth and Wailord as well as a Pokemon with Dig and Dive. On Route 134 you must take the correct path through the currents to the patch of water you can Dive in. Once below folow the path up to the Braille inscription and resurface. Go to the next inscription and use Dig. Make Relicanth the 1st of your party and Wailord the 6th, then read that inscription. Three earthquakes will trigger and open passages on each respective route.
Source 2: My friend entered the Hall of Fame with a Regirock.
Hope I helped!