What should I do about a delayed development project?

I hired a company to build a web project for me. Unfortunately, I didn't specify project completion date and milestones. I was told the project will be complete in 3-4 months and I believed the company president because the company was referred to me through a good referral.
The project is now 2 months behind schedule and I'm 2/3 of the payment through. The web company looks not in a rush to complete my job.

Any suggestions what I should do?

It depends on the contract you have, on the need of having the final product ready, on you having the source code and what kind of post-project relationship you would like to maintain with the provider.

As an overall approach, I would suggest to have a meeting with the most senior management person you can on the provider (maybe the president himself). Explain the situation in details:

1. When the project was kicked off
2. What was your expectation regarding the finish date
3. What results are achieved up to now
4. What results are due
5. What kind of problems you are facing due to the fact you don't yet have the software ready
6. What alternatives do you believe feasible to settle the conflict

If you can't figure out a solution after this meeting, I would suggest you to get a good lawyer and start thinking about more extreme actions such as suing the provider.

Take your contract to a lawyer, and see if there is any way of restructuring the contract. Try to get a due date in there, and a penalty if they do not hit that due date.

It really depends on what you want. Assuming you just want a finished product, I'd set up frequent calls/meetings with the company, even as frequent as daily. I would ask for daily updates listing what was completed today. I would push for frequent deployments to a staging/user acceptance environment so that you can see the progress being made. They may push back on you when you ask for this, but if they are 2 month overdue on a 3 month project, I think you are completely within you rights to expect them to provide you this sort of involvement.

Knowing that you are taking an active role in getting the project finished will most likely make a significant difference.

The other approach is to contact a lawyer and see if you can get your money back and start with someone else. If you really think this contract is dead, then do this, but the pain in this is obvious. This is a startup discussion, so I assume more than anything you want the contracting company to complete the project as quickly as possible.