You should start your fault finding by checking the polarity of your components. It is very important that your LEDs, capacitors and transistors are the right way around because any error will have an impact on the circuit. You can put resistors in any which way and they will still work the same way. Here are some images to help you identify the correct polarity for your components:
Double check you've got the transistors in the correct way as well by comparing them to the diagram. Are they actually NPN transistors? They should be labelled BC548. The kit also includes PNP transistors which will not work.
I've recreated the circuit for you here in this online simulator so that you can better see how the oscillator works. I find it useful to help visualise what's happening in the circuit. I've found that removing either of the transistors, capacitors or the 47k resistors will cause only one of the LEDs to light, so it could be a problem with how you've installed one of those components. Double check and triple check your work and I'm sure you'll find the answer soon enough. If you're still having trouble, try attaching a photo of your circuit and someone might be able to identify the problem.
As for the missing parts, you should definitely bring it to the attention of the store your bought from. The online description of the kit clearly says that "all" components are supplied, so I'm sure they'll fix you up.
Edit: Looks like my link to the circuit simulator didn't work. You can see a YouTube clip of a similar circuit here