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)