Define large.

In Mesoamerica, different shield designs were used at various points in time. Ross Hassig talks about changes in Mesoamerican shield design in his book War and Society in Ancient Mesoamerica.

The Central Mexican city of Teotihuacan, which fell some 500 years before the Aztec empire arose, appears to have used rectangular shields. The most famous depiction of these can be found on the Teotihuacano soldiers depicted in Tikal Stela 31. Given the way this shield is carried, it appears to have been made out of a flexible material and was likely more useful for deflecting glancing blows from projectiles rather than for melee combat. Because the material was likely flexible, it wouldn't have been practical to make them very large. Maya warriors during the Classic period were rarely depicted with shields. Those depictions that do exist typically show small rounded bucklers that are even smaller than these. A comparison of these two types of shields can be seen in this image. (With a caveat, the labels in that image are inaccurate, the different shield types correspond to regional differences not class distinctions). In contrast to the rectangular shields, the rounded shields appear to be rigid (likely made of wood), and were likely designed to deflect blows from clubs.

The rounded shield that you see in most depictions of Aztec soldiers was largely prominent during the Postclassic period (c. 900 AD to conquest). This shield design appears to gain prominence in Mesoamerican armies about the same time as the macuahuitl, a sword-like weapon made of wood and obsidian. The rigid portion of this shield was as large, if not larger, than the metal shields carried by the conquistadors. However, as you can see in the image, it also had a feather fringe on the outside that, like the flexible shields used by Classic Period Central Mexican soldiers, was likely designed to deflect glancing blows from projectiles. If the fringe is considered part of the shield, they were actually quite large.

What you're probably asking though is why they didn't develop the large tower shields like the kinds you frequently see in movies depicting Roman legions. I can only speculate, but it's likely that those kinds of shields were designed for a different kind of combat. As Hassig explains (and as I've tried to explain above), Mesoamerican shields were typically designed to either deflect indirect hits from projectiles and/or for hand-to-hand combat between individual combatants. Tight formations of spearmen (like a phalanx) were not frequently employed in most Mesoamerican armies (with the possible exception of some Central Mexican armies during the Classic Period). If the old adage that "form follows function" holds true, it could be that such a design simply wasn't necessary given the tactics used by Mesoamerican armies. It's also possible that such a shield design would not be as effective with materials other than iron, which the Mesoamericans did not have.

I've written a brief outline as to why the Japanese didn't use shields in this comment.