It should be "The family was left homeless." Family is a singular noun, even though it represents more than one person. "The family got left homeless" doesn't sound natural to me.

In terms of meaning there is no difference. 'were left' is the more standard, formal form. 'got left' is more informal

I think Jade is correct. It should be WAS. And GOT LEFT just doesn't sound right, though I'm not sure why.