Yuffie is a missable character (only in the sense you don't have to find her. However she can be found over and over until you get it right so it isn't one to worry about).
She is obtained from forest areas on the map. You need to run around in any forest you find and hope she appears in a fight. The first forest is just after the Mythril Mines that you enter shortly after leaving Midgar. You will know the mines as you run in to the Turks before leaving them. Just after those mines you will see bushy areas on the map. Yuffie is a random encounter within those.
When she does appear, she is easy to beat. Once she is beaten you have to choose the correct dialog options (do not use the save point). The dialog options are:
Yuffie: "You spikey-headed jerk! One more time, let's go one more time!"
"Not interested"
Yuffie: "You're pretty scared of me, huh?"
"Petrified"
Yuffie: "I'm really gonna leave! Really!"
"Wait a second"
Yuffie: "You want me to go with you?"
"That's right"
Yuffie: "All right! I'll go with you!"
"Let's hurry on"
She will run away and potentially steal from you if you choose the wrong options here. If you do mess up then run around in the forest until she appears again.