Lucrecia’s Crystal Cave
You can travel to Lucrecia’s Cave anytime after you have completed the Underwater Submarine Battle on Disc 2 and received the Shinra Sub vehicle. Take the sub and travel above water over to the peninsula where Costa del Sol is located. Go in behind into the waters on the west side of Costa del Sol and look for a river with one grassy side on the left and a rocky side on the right (shown in the screenshot below).
Submerge the Shinra Sub and you should find a cave nearby (use the screenshot below for reference). Bring the sub back up to the surface when you reach the waters on the other side of the cave and then dock your sub on the rocky ground on the west side of the pool. Use the PHS to add Vincent into your party and then enter the cave - walk forward for the cutscenes to begin.
The game will automatically place your party back outside of the Crystal Cave. After engaging in 10 battles you can return to Lucrecia’s Crystal Cave with Vincent in your party again to retrieve his Ultimate Weapon, Death Penalty, and his Level 4 Limit Break, Chaos.