Can you use dingbat fonts in TextEdit?

Are you in text mode or RTF mode? You will need to be in RTF mode. In the Format menu select Rich Text.

In TextEdit, if I go to the font panel and select a dingbat font (Wingdings, Webdings, Zapf Dingbats) and then type some letters, I just see plain letters. Why won't the special symbols show up?

Unicode systems like OS X normally don't just switch fonts to change characters. You need to use the Character Palette (Edit > Special Characters) to input symbols instead of the keyboard. You can use View = Glyph and choose the font.

Aha! I usually have the character palette set to the code tables view.

Code Tables will also work for Dingbats. They are at 00002700

But Wingdings and Webdings use the Private Use Area starting a E000.

Yes, that's correct. It is sometimes hard to use that area for input, since every codepoint can have multiple glyphs defined for it.