The thing that made me ask this question was that most of Stark family are dead and Bran is now the "heir" to Winterfell. So I guess he should have children to succeed him, otherwise the Stark line will end. (I'm assuming Arya will not come back to Westeros and Rickon's fate is sealed, though I could be wrong.) I don't remember reading anything about it.
In A Game of Thrones (book 1, chapter 25 "Eddard") Ned talks about Bran's future with Arya when she asks if he will ever be a knight (bold emphasis mine):
"No," Ned said. He saw no use in lying to her. "Yet someday he may be the lord of a great holdfast and sit in the king's council. He might raise castles like Brandon the Builder, or sail a ship across the Sunset Sea, or enter your mother's Faith and become the High Septon." But he will never run beside his wolf again, he thought with a sadness too deep for words, or lie with a woman, or hold his own son in his arms.
Ned could definitely be mistaken though, since we aren't explicitly told how he has reached that realization. But given the sad nature of this state of affairs it would be hard to accept that Ned wasn't entirely convinced of it.
There's nothing about Brans injuries that is in any way supernatural, so I think we can safely assume that they resemble those of a real world paraplegic. While many (most?) paraplegic men struggle with impotence, or other dysfunction, the fundamental plumbing does tend to work; after all, blood still circulates to their feet, right?
So, basically, it depends on the specific extent of the injuries he sustained, but there's no reason to think that if someday, he were able to, that it was in any way supernatural or implausible; so long, of course, as he survives that long.
In addition to LessPop's answer there is another thing to consider. (Dance with Dragons spoilers follow:)
Bran is a powerful seer. The one eyed raven is his teacher and I think it's quite logical to conclude that Bran would end up like him. Bran is sitting in a seat all day listening to the One Eyed Raven who is basically part of the trees which seems to imply that one day Bran will become an immovable long lived being who is part of the forest and weirwood. Who knows what insane magical powers such people have? The other thing is, he will probably be presented with choices (like Arya) and maybe there is more than one path.
Since he is most likely the most naturally powerful seer in existence right now (lacking in training), there is a lot open including being killed off. :-) Martin hasn't let us know too much at this stage. It may be that his creatures and whatever else he can create replace any children he would have otherwise had. Also, Bran thinks he can never have a wife. In the world of Westeros that may be true which would mean no "true born" sons.
He is paralysed so the question is does the paralysis extend to his "member" or not. If it is "just" a leg problem, then Bran will be able to conceive children.
Sansa is still alive and kicking, so is Arya, Rickon's tale is still "untold", Jon is alive too ( not trueborn though ) so the Stark line won't die.
I just came across this question and there are a few out of universe answers on spinal cord injuries and fertility. So this is sort of an out/in of universe mix.
Considering that the Game of Thrones is in a time where there is very limited medical advancement, there is a big possibility (medically speaking) that he will not be able to have kids.
Although many are correct that there is a possibility that he may be able to get an erection (typically though this becomes very difficult without the help of modern day drugs such as sildenafil, intracavernosal prostaglandin or mechanical means such as pumps and Electro stimulation, however a Maester may be able to form a pump of some sort to work and even design something that could provide an Electro stimulation) , the semen quality is reduced, most likely due to altered neurogen control of sperm maturation.
The time window for getting normal semen is considered to be from one to two weeks only after injury, after that the quality is greatly reduced. So in Bran's case he has passed that window. Some people have Necrospermia or spermatogenic which will need to use of in vitro fertilization, obviously not available. And without going into too much more detail there can be issues of retrograde ejaculation.
So in summary, even if the delivery method is overcome , which in itsself is a very big hurdle ,there are a host of other issues that relate specifically to the quality of sperm in spinal injury cases, that a very high percentage of people will have to battle. Typically in modern medicine it requires a team approach from several different specialist to get to the ultimate goal of conception.
So in my opinion I would say it highly unlikely that Bran was to ever conceive children using traditional methods for the time. However, there could be some magical potion or spell that could allow this, but that is speculation.
Obviously this is a very abbreviated answer, as whole papers have been written on the topic, so please do not take as medical advise or reference (On a SciFi forum, I would hope not)
Well, lets think about it. His testicles a healthy as according to the wiki of ice and fire he becomes "infatuated with meera" so the testosterone is moving. And therefore he can bear a child. Also if the blood is ticking in the crippled part of his body the plumbing that causes the long and hard should work too. But when it comes to sex he can never be on top. Never have an orgasm or anything else presuming his crotch has suffered the same damage as his legs. Not to mention it would be hard to find a wife as its not an attractive match (its a bit like the Tyrion business, he might look slightly better than Tyrion, but never the less).