I feel that this web application is like a ticking time bomb. The project manager has ignored the project but the CEO has placed high hopes on it.
What should I do?
You should share and discuss your feelings with your immediate manager. If your manager is the Porject Manager, then this is the person to speak to. Your assumption that this is being ignored by the Project Manager may not be accurate. He/She most likely has much more information about the organization's plans and priorities.
Since this is a small company, I would imagine that there may be a few occasions when you can speak directly with the CEO. In a non-aggressive way, you should bring up the topic and share your eagerness.
Speaking behind-the-scenes with another manager or anyone else really is not a good idea. Your manager is your key to prosperity and you certainly do not want to soil this relationship.
Good luck!