We present the results of a pilot survey for neutral hydrogen (HI) 21 cm absorption in the Arecibo Legacy Fast Arecibo L-Band Feed Array (ALFALFA) Survey. This project is a wide-area "blind" search for HI absorption in the local universe, spanning -650 km/s < cz = 2x10^20 cm^-2) is Delta z = 7.0 (129 objects, assuming T_s = 100 K and covering fraction unity); for super-DLAs (N_HI >= 2x10^21 cm^-2) it is Delta z= 128.2 (2353 objects). We re-detect the intrinsic HI absorption line in UGC 6081 but detect no intervening absorption line systems. We compute a 95% confidence upper limit on the column density frequency distribution function f(N_HI,X) spanning four orders of magnitude in column density, 10^19 (T_s/100 K)(1/f) cm^-2 < N_HI < 10^23 (T_s/100 K)(1/f) cm^-2, that is consistent with previous redshifted optical damped Ly alpha surveys and the aggregate HI 21 cm emission in the local universe. The detection rate is in agreement with extant observations. This pilot survey suggests that an absorption line search of the complete ALFALFA survey --- or any higher redshift, larger bandwidth, or more sensitive survey, such as those planned for Square Kilometer Array pathfinders or a low frequency lunar array --- will either make numerous detections or will set a strong statistical lower limit on the typical spin temperature of neutral hydrogen gas.