Do larger multicellular organisms have an increased risk of mutation and thus cancer?
As mentioned in the comments, this question is quite complicated. If the chance of a single cell from different organisms getting cancer was the same, then you would be correct, but this is not the case.

Different organisms have evolved to live different lengths of time. This is rather obvious when you think about it: mice have a maximum lifespan of ~3 years, humans ~80years (these figures are of course massive generalizations - but it doesn't really matter). Yet 'old' mice (> 2 years) still get age-related diseases such as cardiovascular disease, and cancer. It should be fairly clear that the risk of a 2 year old mouse getting (age-related) cancer is much higher than a 2 year old human.

The explanation for this isn't exactly concrete, but it goes something like this; mice invest a lot more resources in growing very fast and reaching a reproductive age very early, whereas humans develop much more slowly, and invest a higher proportion of their resources in maintaining tissue function, thereby reaching reproductive maturity much later than the mouse. The Hayflick limit for a mouse is about 10, whereas for humans this is closer to 60.

Thus the rate of aging in humans is much lower than that of the mouse, and it comes back to evolution: mortality in the wild is very high for the mouse, and thus mice that invest heavily in reaching early reproductive maturity are selected for - and those that invest in anti-cancer mechanisms take longer to reach reproductive maturity, and thus have less chance of actually doing it!

So to answer your question - smaller organisms tend to have higher risks of getting cancer because their lifespans are comparatively shorter, and thus they 'age' at a faster rate and invest less resources in maintaining function (e.g. genome integrity).