Me and my wife recently got married and the father of her 3 year old hasn't been around and we have tried to make contact with him about seeing his child but never came around.. can I adopt the baby girl because she doesn't carry his last name?

You can adopt the child even if she has his last name, that has nothing to do about it.

You hire an attorney, ( if he has not been proved the father in court you will have to get a DNA test also) but he will have to sign over his rights, this is part of the adoption process.

Normally they will be glad if they don't want the child, since when they sign over the rights their child support stops

First, please pay more attention to posting guidelines. There is a Read First sticky at the top of the Adoption forum (where this was moved from) which directs questions of a legal nature here.

Whether he signed the birth certificate or whether the child bears his name is immaterial. To adopt a stepchild you have to get the bio parent to relinquish their rights.