I'm the grad student mentioned above by Kevin Costo. I should give a little disclaimer, which is that I am self-diagnosed based on the VVIT questionnaire. I was reading an article that described aphantasia, and instantly recognized myself in the description.
As Kevin mentioned, I do geometric topology. I find that it doesn't present much of a problem for me, pretty much because I can still keep track of relationships between, say, points - I just don't have a mental image. As an example, in the opening of Thurstons book on 3- manifolds, he talks about what one would see when sitting in a 3-torus and looking around. This, while "visual," made perfect sense to me.
Another example is the game Skribble, in which one person decribes a drawing of a shape (say, a train) using simple geometric shapes (rectangles, circles, etc). The idea is to guess what the shape being described is. I can play this game in my head. How? I don't know, to be honest. I know I don't have a picture in my head, but I sort of know the schematics, how things relate to each other.
I'd say the biggest problem I've run into is when something is sufficiently complicated and I need a sequence of steps to simplify it. If I need to, I just draw a picture. In some ways, I think it may be easier for me to be a geometer than, say, a category theorist, because I've already found ways to think about actual objects and their relationships. I don't have a way to visualize abstract objects.