We report simultaneous multicolor near-infrared (NIR) observations of the supernova associated with X-ray flash 060218 during the first 16 days after the high-energy event. We find that the light curve rises and peaks relatively fast compared to other Type Ic supernovae (SNe Ic), with the characteristic broad NIR peak seen in all three bands. We find that the rise profile before the peak is largely independent of NIR wavelength, each band appearing to transition into a plateau phase around day 10-13. Since the light curve is in the plateau phase when our observations end at day 16, we can only place limits on the peak absolute magnitudes, but we estimate that SN 2006aj is one of the lowest NIR luminosity X-ray flash/gamma-ray burst (XRF/GRB) associated SNe observed to date. The broad peaks observed in the JHK$_s$ bands point to a large increase in the NIR contribution of the total flux output from days 10-16. This evolution can be seen in the broad color and spectral energy distribution diagrams constructed using UBVRIJHK$_s$ monochromatic flux measurements for the first 16 days of the event. Ultimately, a 10 day rise time would make SN 2006aj an extremely fast rise SN Ic event, faster than SN 1998bw and SN 2003dh, which combined with its underluminous nature indicates a lower amount of $^56$Ni ejected by the progenitor compared to other XRF/GRB-SNe. Furthermore, the lack of significant color change during the rise portion of the burst points to little or no spectral evolution over the first 10 days of activity in the NIR.