Here's the thing: Charlotte is a spider, and spiders don't use ink or pens! They spin webs to catch their food and for shelter.
What Charlotte *does* is write messages in her web using the silk strands themselves. She carefully manipulates the silk to form letters, which is how she creates words like "some pig" and "humble."
So, Charlotte doesn't write on her web *like* we write on paper, but she does use her web as a canvas to create messages for others to read.