Carry a notebook so you can jot down ideas from your own life. Tune into your surroundings and look for phrases or words you can use as a punch line in your cartoon story. Also, notice how different types of people around you speak.
Brainstorm or use a word web to get ideas on paper. An effective word web would be to come up with a general character type such as a baker. Write down words that describe what the baker does, where, the types of tools and raw materials used and the language a baker would use. Review your notes for a possible cartoon story, punchline, scene descriptions or dialogue.
Use the "cascade writing technique" by writing the dialogue for your first cartoon panel at the top of a sheet of paper. Underneath, write the dialogue for the next panel. Continue to write in this fashion. When you are done, analyze the dialogue for possible jokes, double word meanings, or plot misdirections.
Flesh out your cartoon story using the dialogue from your cascade writing exercise. Use your word web and brainstorming ideas to describe the panels for the illustrator. Create tension within your cartoon, and then release that tension with the punch line you developed.