With the following link, you can create a custom one-character font. If you create the one character as a vector drawing of a chili pepper, you could type the character at the end of the line and apply the character style. You could also have the character style applied with a grep style or a nested style.
Migintosh's method will work but it's not necessary. (I'd use that method if I had to do this repeatedly on many projects.)
Simpler is to use Inline Graphics. You can read about them in this Help article:
Here's an example like you're creating by pasting a chili graphic into a tab position in the menu:
You also can move an object to a position in a layout and make it an Inline Graphic. When the object is selected, a small blue square appears at the upper right. If you drag the blue square, it anchors it. But what you want is to make it Inline (placed at the baseline within the text). So hold down the Shift key while dragging it into position.
I'd do it that way too, Steve, but the OP asked about character styles, so I assumed he wanted to use type.
Sorry for the late response. Thank you both, Migintosh and Steve. I have tried both methods and they accomplish what I was looking for.