TL;DR: generation of high power microwaves from organic mechanisms is very, very hard and evolutionarily unlikely. Microwaves are also a terrible way of tunnelling through ice. Note that polar bears and leopard seals don't bother in the real world; who are you to argue with those highly effective apex predators?
As another answer already touches on the inefficiency of melting ice (even more so when you consider that there's no energy-dense food in there to make the tunnelling worthwhile!) I won't go into that, but lets consider the physiological issues of microwave generation.
Microwave generators have two important kinds of material that are hard to come by in nature: microwave-transparent materials, to cover and protect the business end of the emitting device, and microwave-reflecting materials to ensure that all the microwaves go where they are wanted (eg. out the front) and aren't absorbed by the rest of the equipment which would cause undesirable power loss, heating and damage.
Lightweight polymers like polystyrene or polythene are easy to come by courtesy of modern chemistry and industry, but there aren't many good biological equivalents. The presence of any water is undesirable (because of the resulting energy loss, heat generation and ultimately physical damage that will result) and growing a water-free biological membrane sounds like an absolute nightmare, and something deeply unlikely to evolve naturally. It is remotely possible that you could have some sort of super-thin layer of specialised bone, horn, hair or tooth and have the extruding layer of cells die off, maybe.
Similarly, sheet metals and metal meshes have been available to humans for millenia, but bulk deposition of metals in organic creatures isn't a thing, let alone formation of nice smooth reflecting cavities. There's no obvious evolutionary path to such a trait... seems like mineral/protein composites that most tooth or bone or shell-bearing animals produce is quite good enough without the hassle and energetic expense of refining metals.
Finally, even if you've solved those two not-insignificant problems, you need to actually generate microwaves. Human approaches to this involve vacuum tubes (good luck evolving a vacuum-sealed cavity in a biological organism!) or semiconductors. The latter might just possibly be able to have some organic counterparts but: they are low power microwave sources, the sort useful for communication, perhaps, but not for generating the enormous amounts of power you need to melt enough ice to fit your body through.