Catch-able through normal gameplay:
* Raikou, Entei, and Suicune: These legendary beasts are roaming around Johto after you beat the first gym leader. You can track them down by listening for their cry using the Radio Tower in Ecruteak City. This is a common and popular strategy for capturing them.
* Lugia: You can catch Lugia in the Whirl Islands after defeating the Elite Four, but before facing the champion, Lance. This means you can technically catch Lugia before your "real" championship.
Catch-able with specific items/events:
* Mewtwo: You can catch Mewtwo in the Cerulean Cave after defeating the Elite Four. You'll need to have obtained the Silver Wing, which requires doing the "Mewtwo event" in the Kanto region.
Uncatch-able before the league:
* Articuno, Zapdos, and Moltres: You can't catch these legendaries until after you become champion, though they are in the game and can be found in the Kanto region.
* Ho-Oh: You can catch Ho-Oh after you become champion, but only by triggering the "Ho-Oh event" in the Kanto region.
* Groudon and Kyogre: These legendaries are not available in SoulSilver until you have completed the game and have access to the "Emerald storyline" using the Pokewalker.
Important Note: The Silver Wing, needed to catch Mewtwo, requires you to travel to Kanto, which is only accessible after defeating the Elite Four. This makes Mewtwo essentially uncatch-able before the league, even though you can enter the Cerulean Cave earlier.
So, in summary, the only legendary Pokemon you can realistically catch before the Elite Four in SoulSilver are Raikou, Entei, Suicune, and potentially Lugia (depending on how you define "before").