I'm assuming that you have your textbox set to 'multi-line' ? ( That is done in the properties window, on the right) If your using a word wrap, once the textbox area is filled, the scrollbar should pop in on it's own if you have the vertical scrollbar set to auto. If you set the horizonal scrollbar to 'auto', then the text will pan right, you DON'T want that! I usually use a RichTextBox to get the auto-scroll, I rarely use a standard textbox for that.
I'm not on my design computer at this time, or I could get a lot more in-depth here. I might add to this when I get home, on my design computer.
I'm home.
To add a scrollbar to your multi-line textbox, just go to the 'Properties' window & select Vertical in the scrollbars line. Your options are None, Vertical, Horizonal, or Both. You want 'Vertical'. It's just that easy. As you type past the parameters of the textbox, the scrollbar will automatically pop in, & start working! That's Easy!!! :-)
There ya go! Good luck, & Happy coding to you!