Even if a company has no posted job openings, that doesn't mean you can't get a job there. Sometimes you can get your foot in the door by impressing someone in customer service.
Photo by CWCS Managed Hosting
If you have a great idea to help a company serve its customers better, customer service is right there with you. The Muse suggests this approach might be a different way of getting your foot in the door at a company:
Start by expressing your loyalty to the company (think: You've always admired this startup, which is why you feel so strongly that a particular product or process should be improved). Then, request to be connected with an appropriate person to discuss further, but make no mention of your career interests whatsoever.
Once you get to the right person and explain your suggestions, the conversation of employment might come up. That's when you mention that you'd love to work there. If it doesn't come up, you already have a contact at the company. Now that you know the name of a person in that department, you can send them your resume and a cover letter following up on the conversation.