Bella becomes pregnant in Breaking Dawn, the fourth book of the Twilight saga. While Edward and Bella are on their honeymoon after they got married, Bella and Edward... "did it" and she became pregnant, like any other human. It is possible and here is why: Bella mentions the myths she read online when she first discovered Edward was a vampire. The word INCUBUS comes to mind. The definition of incubus is: the ability to father children with his hapless pray. From more detailed definitions, it leads to this: certain people (in this case, vampires) have the DESIRE to be able to produce children. So like in the passage below from the book Breaking Dawn, it is obvious that Edward wishes to have that ability. Well, lucky for him, he does. But it doesn’t mean all vampires do. It’s kind of like his mind reading. It was a gift chosen for him, not all vampires have it. This also explains why Edward has the ability to get Bella pregnant even though he is not human.
Passage-
(Some stuff about Edward making sure Bella’s okay with staying eighteen forever.)
“Never changing… never moving forward.” (Edward)
“What does that mean?” (Bella)
He answered slowly. “Do you remember when we told Charlie we were getting married? And he thought you were… pregnant?" (Edward)
“And he thought about shooting you,” I guessed with a laugh. “Admit it-for one second, he honestly considered it." (Bella)
He didn’t answer.
“What, Edward?” (Bella)
“I just wish…well, I wish that he’d been right.” (Edward)
“Gah,” I gasped. (Bella)
“More that there was some way he could have been. That we had that kind of potential. I hate taking that away from you, too.” (Edward)
It took me a minute. “I know what I’m doing.” (Bella)
“How could you know that, Bella? Look at my mother, look at my sister. It’s not as easy a sacrifice as you imagine.” (Edward)
“Esme and Rosalie get by just fine. If it’s a problem later, we can do what Esme did-we’ll adopt.” (Bella)
He sighed, and then his voice was fierce. “It’s not right! I don’t want you to have to make sacrifices for me. I want to give you things, not take them away from you. I don’t want to steal your future." (Edward)
But because Edward wants and option of having a child he has the ability to make one. That's what happened to Bella and Edward, that's how they made Renesmee.
If Edward never wanted the option to get Bella prego then it would have never happened. It was his choice. Edward didn't know he had a choice though or he would have never agreed to "try."
While Bella is pregnant Edward is begging to get the baby out of Bella but she keeps telling him "No," Bella loves the baby. But the baby is killing Bella, the baby is drinking all of Bella's blood because it is half vampire (explaination below).
And-incubi (Edward) have their own special form of sperm depending on what kind of a creature they are. For Edward, it just happened to be venom since he is a vampire. There are legends and stuff about horse blood being the sperm therefore the baby is half horse. So if Edward’s sperm is vampire venom, then clearly the baby will end up half vampire and half human.
When Bella is giving birth it is too hard on a human and Edward had to bite Bella so she could live.