Sure you can. Swarms!
Have you ever seen the Carolina Bays?
Swarms of meteor strikes on land would leave swarms of craters. The Tapanui craters don't seem nearly sufficient enough to create a century of cooling. Moreover, the "Mini Ice Age" wasn't until the 14th century, well removed from the alleged timing of this impact.
I read the full article and lemme say that I'm a bit skeptical. There is a curious amalgam of oceanic and atmospheric features that don't seem altogether coherent. There should be additional signs of widespread meteor strikes that are missing (why did the shower of impacts miss China, or Europe?) or ambiguous (astronomers quite clearly knew the difference between comets and meteors).
I have no problem with the idea of tsunamis afflicting the Amerindians and altering the course of their civilizations. I would think seaquakes are a perfectly fine explanation. Altering the El Niño phenomenon would require a major disruption of the ocean-atmosphere dynamics of the Tropics; I don't see how relatively mild (in the grand scheme of things) meteor strikes accomplish the task.
Nevertheless, I think that there is a decent case to be made for meteor strikes in the late 12th century Pacific Basin, but I think this thesis falls apart on account of overreaching with an extrapolation back to Japan, China, and Mongolia that seems tenuous, at very best.
PS. I could be mistaken about this, but I thought the evidence of a historical tsunami can be discerned from the geology of a region. Is there actually geological evidence in Peru or Mexico of an enormous 12th century tsunami?