Can oil be replaced with yoghurt in a cake recipe?
Short answer - not really. Fat is an essential component in any cake, and milk just isn't very fatty - about 5% for whole milk. You can make cakes with milk, but they require totally different recipes: you can't simply substitute milk for oil.

Bear in mind that you're distributing the cup of oil throughout a whole cake, so that any one slice will only have a fraction of the oil. I assume you'll also be sharing the cake with others, so you'll be 'spreading the calories' somewhat.

You should be using a neutral oil like canola (rapeseed) oil anyway, as it has a relatively mild flavour. A popular alternative these days is to substitute all but a couple of tablespoons of the oil with apple sauce (really), but this can be hit and miss and doesn't work with all recipes - it usually works best with things like quick breads (scones etc).

Of course, there's also the point that cakes are supposed to be a treat, and they'd be less of a treat without the fat, in which case why bother? If you want to be healthy, make a salad. If you want a treat, make a cake and don't worry about the contents too much.