We present theoretical calculations for ohmic contact technology to wurtzite Silicon Carbide using thin Indium Gallium Nitride and Aluminium Indium Nitride cap layers. Spontaneous and piezoelectric polarization in Indium Gallium Nitride and Aluminium Indium Nitride cap layers gives rise to bound interface sheet charge density of the order of 1013 electrons per cm2, and built-in electric fields of the order of MV per cm. For Si-face p-type SiC, the large compressive strain in very thin InGaN and AlInN cap layers results in negative sheet charge densities and much lower tunneling widths for holes compared to bulk contacts. For C-face p-type SiC, pseudomorphic nitride layers yield no benefit over bulk contacts since positive interface sheet charge densities repel holes and give higher contact resistances. However, thick, relaxed cap layers lead to spontaneous polarization-based negative charge densities that attract holes, leading to lower contact resistances. The contributions of the heavy, light and split-off band holes to the total tunneling flux is taken into account in our calculations. All tunneling probabilities are calculated within the Wentzels-Kramers-Brillouin approximation. The presence of the appropriate cap layers leads to several orders of magnitude improvement in tunneling transmission probabilities, and in many cases, makes ohmic technology to SiC feasible. While the major difficulties are encountered for contacts to p-type SiC, the effect of cap-based polarization technology on contacts to n-type layers is also discussed in this paper. Our calculations have relevance to contact technology for bipolar devices built from wide band-gap wurtzite semiconductors like heterojunction bipolar transistors, light-emitting diodes and lasers.