How to get Vincent and Yuffie

Vincent is a more time-consuming character to get than Yuffie, but he is well worth it. You can get Vincent as soon as you get to Nibelheim after finishing the Cave of the Gi. You need to fight a challenging boss in order to get Vincent, easily one of the harder ones you've fought so far so make sure to stock up on whatever you need.

When you get to Nibelheim, enter the Shinra mansion located north of town. Climb the staircase and go to the left hallway. There is a safe inside the room that's above you. You can find hints to the code around the mansion, but for the purposes of this guide the code is below:

Right 36, Left 10, Right 59, Right 97

You have to make sure you only move in the direction stated. If, let's say, you accidentally pass 36 and land on 37, you have to restart again. After you've opened the safe you'll find a key (necessary to get Vincent), the Odin materia and Red XIII's Ultimate Materia, Cosmo Memory, which does massive damage (nearly 8 times normal damage). After you've opened the safe, you'll immediately be thrown into a boss battle with Lost Number, most likely the hardest battle you've fought yet, especially if you're unprepared and don't know how Lost Number fights.

The battle with Lost Number will be a tough one. Lost Number consists of two halves; the purple physical half which performs physical attacks and the red magic half which performs magic attacks. After reducing his HP below 3,498, Lost Number will change into one of his sides, depending on whether the hit that made his health go under 3,498 was magical or physical. The physical side is MUCH harder than the magic side. As such, it's recommended to use sense and monitor the HP so that it only goes under 3,498 with a magic attack. If you have Big Guard (which you should, it's an extremely useful skill which casts Barrier, M.Barrier and Haste on your entire team) use it immediately. You can get Big Guard from Beachplugs which are found around the beaches in Costa Del Sol and Golden Saucer. You should also use White Wind if you have it to constantly keep the party healed. After it transforms into its magical form, its defense decreases substantially and its M.Defense and Magic will increase. You should take it out with physical attacks at this point, as magic won't do much damage. It's recommended to use Aqualung and Beta at the start, and finish it off with limits after it transforms. Of note, Aeris's Seal Evil and Cloud's Cross-Slash limits will stun Lost Number for a short while.
After beating Lost Number make sure to pick the key and Odin Materia. With the key, you can now get Vincent! Head to the basement and open the previously locked door. Inside, you'll see various coffins, but there's one in particular that's different from the rest. Interact with it, and Vincent will come out. Talk to Vincent, name him and when you're about to leave the basement, Vincent will stop you and join the party.

Yuffie is the easier character to get. She's a ninja/thief from the land of Wutai, and you can get her as early as Junon! Once she gets her ultimate weapon, Conformer, she becomes extremely useful in morphing and can be used to max a character's skill to 255 without using any sort of cheats.

You can meet Yuffie in any forest. You can meet her as early as Junon, but the forests around Junon have the lowest chance for her to show up. Get crackin'.

Thanks for reading my guide, hope it was helpful! If you need any more hints/tips, let me know. I'll try to get back as soon as I can.