did they have market places?
Archaeologists have found traces what could have been formal arcade style masonry architectures and foundations of market stalls; and the Spanish colonialists have written about encountering a thriving market economy.
Maya civilization
Spanish reported a thriving market economy when they arrived in the region. At some Classic period cities, archaeologists have tentatively identified formal arcade-style masonry architecture and parallel alignments of scattered stones as the permanent foundations of market stalls. 
how far back can we trace the mayan culture As far as circa 2000 BC what period of history is that? The Archaic period where were they located? The civilization encompassed areas of south-eastern Mexico, all of Guatemala and Belize, and the western portions of Honduras and El Salvador. did they practice politics? Yes, there were states and chiefdoms which had fluctuating policies towards one another. did they have societies? The society was hierarchical with a sharp difference between the elite and the commoners. did they fight with each other? Yes, clashes were common between states and chiefdoms. what kind of warriors were they? Well trained, and in the Contact period, their armies were highly disciplined where warriors participated in regular training exercises and drills. did they practice trade as well? Yes, and control of trade routes or access to important trade goods contributed to the growth of cities. describe their merchants? Not much is known about them, but it seems that at least some were members of the elite who engaged in long distance trade, while the majority belonged to the middle class largely involved in local or regional trade.
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Maya civilization [SEP] Marketplaces
Marketplaces are difficult to identify archaeologically. However, the Spanish reported a thriving market economy when they arrived in the region. At some Classic period cities, archaeologists have tentatively identified formal arcade-style masonry architecture and parallel alignments of scattered stones as the permanent foundations of market stalls. A 2007 study analysed soils from a modern Guatemalan market and compared the results with those obtained from analysis at a proposed ancient market at Chunchucmil. Unusually high levels of zinc and phosphorus at both sites indicated similar food production and vegetable sales activity. The calculated density of market stalls at Chunchucmil strongly suggests that a thriving market economy already existed in the Early Classic.