We present the results of a pilot survey for neutral hydrogen (H I) 21 cm absorption in the Arecibo Legacy Fast Arecibo L-Band Feed Array (ALFALFA) Survey. This project is a wide-area "blind" search for H I absorption in the local universe, spanning -650 km s-1 < cz < 17, 500 km s-1 and covering 517.0 deg2 (7% of the full ALFALFA survey). The survey is sensitive to H I absorption lines stronger than 7.7 mJy (8983 radio sources) and is 90% complete for lines stronger than 11.0 mJy (7296 sources). The total redshift interval sensitive to all damped Lyalpha (DLA) systems (N_{H\,\mathsc{i}} \ge 2\times 10^{20} cm-2) is Deltaz = 7.0 (129 objects, assuming Ts = 100 K and covering fraction unity); for super-DLAs (N_{H\,\mathsc{i}} \ge 2\times 10^{21} cm-2) it is Deltaz = 128.2 (2353 objects). We re-detect the intrinsic H I 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_{H\,\mathsc{i}},X) spanning four orders of magnitude in column density, 1019 (Ts /100 K) (1/f) cm^{-2} < N_{H\,\mathsc{i}} < 10^{23}\ (T_s/100 \,K)\,(1/f) cm-2, that is consistent with previous redshifted optical DLA surveys and the aggregate H I 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.