Can I make the front camera the default camera?

It doesn't surprise me that the stock camera app won't let you do anything when there is no rear camera installed. I once wasn't able to boot an iMac because the EyeSight camera was disconnected.

It's likely that the flash is shorted out, in which case you won't be able to use your existing unit. It could be worth your while to clean the contacts with a high concentration alcohol and cotton swab, but I doubt that will do anything. If you want a cheap option, try downloading a third party camera app like Camera+ and see if it allows you to select the front camera.

If none of that works, you'll just have to get a new rear camera. It's a pretty major component of the iPhone, so not having one doesn't really seem worth it.