Why would Bran Stark not have a wheelchair? Doran Martell uses one due to his gout, so one would think that a comparable 'technology' would exist in the North. Do we have a sense of whether the terrain in Dorne is just much more wheelchair friendly?
Wheelchairs are impractical if you don't have paved roads and ramps or elevators instead of stairs everywhere. Possible reasons why it works for Doran Martell:
- He seems to be content to spend his time thinking and watching, not moving around much. Bran wants to be active.
- Chronic gout does not make you completely unable to walk (as opposed to paraplegia): the muscles work and can be controlled, but the joints are damaged and aching. With the help of someone to lean on, Doran is probably able to walk short distances and use stairs.
- Gout is a gradually worsening condition, so there was time for him to have buildings modified, or to move to more wheelchair-friendly ones. The book states that he spends most of his time at the Water Gardens, a palace that is described as mostly level ground, quite unlike Winterfell.
Futhermore, wheelchairs are probably not widely known at all - Commoners couldn't afford them, Doran's was constructed especially for him, and he explicitly avoids being seen in it since he does not want his enemies to seek him weak. So it's quite possible that nobody in the North has even heard of the concept.
Also, consider that he could have used the wheelchair while he was in Winterfell, but when he got on the road with Hodor and co, and went north of the Wall, where roads aren't so good as they are south of it, it would be really hard to push the wheelchair around.
You can see something similar in King's The Dark Tower series, where Susannah has a wheelchair, but it is really hard for her companions to push her, especially when they are in a forest or a rocky terrain etc.