Absolutely no chance. I don’t care about swearing, and the gross out vomit stuff is not a problem, but a 7 year old is definitely not old enough to understand the context of the behaviour in the show.
A lot of the humour is based on the outrageous transgression of social conventions and norms. At that age I doubt you’d understand that. There’s a lot of very bad behaviour in it, not to mention taking the piss out of black and disabled people, and I would hate to think that my kid would be seeing it on telly and thinking it’s normal.
I remember even as a late teenager repeating some crappy things that I’d seen on telly in comedy shows (Blackadder for one) and it led to some uncomfortable situations. I was old enough to realise the difference between telly and real life, but only AFTER it’d come out of my mouth.
Kids that age are still picking up a lot of their social codes from what’s around them, and that includes TV. Family guy is a PARODY of society, and you have to understand something fully before you can understand a parody of it.