We present the early-time light curves of Type Ia supernovae (SNeIa) observed in the first six sectors of Transiting Exoplanet Survey Satellite (TESS) data. Ten of these SNe were discovered by ASAS-SN, seven by ATLAS, six by ZTF, and one by Gaia. For nine of these objects with sufficient dynamic range (>3.0mag from detection to peak), we fit power-law models and searched for signatures of companion stars. We found a diversity of early-time light-curve shapes, although most of our sources are consistent with fireball models where the flux increases as {prop}t2. Three SNe displayed a flatter rise with flux {prop}t. We did not find any obvious evidence for additional structures, such as multiple power-law components, in the early rising light curves. For assumptions about the SN properties and the observer viewing angle (ejecta mass of 1.4M{sun}, expansion velocity of 104km/s, opacity of 0.2cm^2^/g, and viewing angle of 45{deg}) and a further assumption that any companion stars would be in Roche lobe overflow, it is possible to place upper limits on the radii of any companion stars. Six of the nine SNe had complete coverage of the early-time light curves, and we placed upper limits on the radii of companion stars of <~32R{sun} for these SNe, <~20R{sun} for five of the six, and <~4R{sun} for two of the six. The small sample size did not allow us to put limits on the occurrence rate of companion stars in the progenitors of SNe Ia. However, we expect that TESS observed enough SNe in its two-year primary mission (26 sectors) to either detect the signature of a large companion (R>20R{sun}) or constrain the occurrence rate of such systems, at least for the fiducial SN properties adopted here. We also show that TESS is capable of detecting emission from a 1R{sun} companion for an SN Ia within 50Mpc and has a reasonable chance of doing so after about six years.