I recently changed my iPhone 4's LCD + screen and had the same problem but with rear (big) camera. It all works great (including front facing camera) except when I open camera app shutter stays closed until I press the button to change camera, then front facing camera image will appear.
On other posts I read talk about a small connector that easily falls off during reparation, I still got it. Another post suggested the two black prongs coming from the camera block were its power connectors, I noticed they were coated in black, which doesn't look very conductive...
I also bought a new rear camera and that wouldn't work either. The last thing I could think of was there being contamination in the connector that wouldn't let it contact so I proceeded with a delicate cleaning of all the connectors ant took the logic board back out to make sure there was nothing strange going on under there.
At this point I started thinking it might be a software related so I did a Settings reset: same, Full restore: same, tried third party apps: same.
What's the next step?
(This is not my first iDevice (iPods, iPhones, Macs) fix, but the first iPhone 4 with this problem. I am was very delicate during replacement (first unplug battery, plastic tools(as much as possible)... and am sure all is in its place.)