Your sentence with 'using' is technically incorrect because of the comma that precedes 'using'.
(It appears) that natural rubber was first processed (by) the ancient Mayan people**,** (using) it to make sandals, figurines, and even some rundimentary sports equipment. (No error)
has the meaning of
It appears that natural rubber was, using it to make sandals, figurines and even some rudimentary sports equipment, first processed by the ancient Mayan people.
This clealy does not make any sense.
However, omitting the comma changes the meaning of the sentence
It appears that natural rubber was first processed (by) the ancient Mayan people [using it to make sandals, figurines, and even some rundimentary sports equipment].
Here, the rest of the sentence following 'using' is an integral part of the noun 'The ancient Mayan people' This additional information specifies exactly which Mayan people you are referring to.
It appears that natural rubber was first processed by the ancient Mayan people using it to make sandals, figurines, and even some rudimentary sports equipment. (And not the ancient Mayan people using it to make jet fighter planes, Ak-47s and even rudimentary aircraft carriers).
Of course, this is a ridiculous distinction to make, which is why the variation provided by Peter Shor, 'who used it to make...' sounds far better to a native English speaker.
But again, just to be clear, technically the original sentence was incorrect because of a tiny little comma between 'Mayan people' and 'using', which removed the 'using...' part of the sentence from defining the Mayans, and instead made it function adverbially with the verb 'was'. Of course in spoken English, it is virtually impossible to hear this comma and the 'correct' meaning of the sentence based on context is usually heard.
This interpretation of the sentence is based largely on Michael Halliday's 'Systemic Functional Grammar', which may or may not have any bearing on what the SAT exam uses. An overview of SFG's 'embedded clauses' (using it to make...) can be found On the last video on this page. Google Books has preview pages of Halliday's full work here.