My sister went to an engagement party over the weekend and she called me the morning of the party to ask an etiquette question: Would it be okay to bring her baby to the party?
My first response was what happened to the babysitter. Nothing they (his parents) were still planning on babysitting.
My second response was to ask if the baby was invited to the party to begin with. She wasn't sure. So I asked if the envelope the invitation came in read : Mr. & Mrs or The Smith Family? She hesitated but said just to the two of them. So I told her I didn't think it would be appropriate to bring the baby. It was clear to me that she wasn't invited.
She asked me if there was any way she could bring the baby and I told her that she could call and ask the host if it was okay but since the reason wasn't that the babysitter canceled at the last minute she would have to just ask nicely and hope that her friend said it was okay. Which she did.
If you get a call from a guest who at the last minute wants to bring their kids it's up to you to make the cal based on space, money and your relationship to the guests.
In some instances there physically will be no more room at the site. If the kids are over the age of 4 their parents may expect you to cover their meal - that will cost you money and you also have to find out if the caterer can do a kids meal at the last minute. Is your wedding kid-friendly? Your guests won't know in advance but you will - think about whether there will be other kids there and what entertainment will be there for the kids.
In the end it's your call and be prepared for the parents to say that if the kids can't come they can't make it. I would hope this won't ruin your relationship but you know your guests better than me so make sure that you are comfortable with your decision - financially as well as emotionally.