There was some small-scale use of Spanish arms and horses by indigenous peoples during the 17th century. However, trade in horses and arms with them was prohibited by the Spanish, so only a small trickle of armour, swords, and firearms made their way into indigenous hands, by illegal trade, theft, or as war booty during most of the 17th century. Peoples under Spanish rule, with their ownership of horses prohibited, could not engage in horse breeding on any significant scale.
The first significant military use of Spanish arms and horses by native forces appears to be the use of horses by Spanish allies in the Mixton Wars of 1540-1542. Spanish manpower shortages resulted in the Spanish allowing the native allies to use horses in the war. The much longer Chichimeca War which followed (1550-1590) saw the use of horses against the Spanish. The Chichimeca captured horses when they could, and used them to provide mobility for their raiders. If they encountered Spanish horses, but couldn't steal them for their own use, they would attempt to kill them to deprive the Spanish of them. They used Spanish weapons which they captured, and are known to have used small numbers of swords in the war. They are known to have captured Spanish guns, and are likely to have made what use their limited supply of ammunition allowed.
Horse ownership in native hands exploded in 1680, when the Pueblo Indians revolted against Spanish rule, with spectacular success, acquiring thousands of horses and the freedom to breed horses free of Spanish interferences. Many of these horses were traded to their neighbours, such as the Ute, Apache, and Comanches, who then proceeded to breed their own. The Comanche established their own substantial empire based on their military use of the horse.
The Spanish return 12 years later was facilitated by Pueblo willingness to work with the Spanish against the threat of raiding by their now-mounted neighbours. The Spanish, with only small numbers, realised that it was mutual - they needed Pueblo assistance against raids themselves. They lifted the previous bans on native ownership of horses and guns.
Unlike horses, guns, swords, and armour needed to be made rather than bred, and supply was limited until large-scale gun trading came. This meant that while guns, swords, and armour saw limited native use in the 17th century, it could only be quite limited use. The arrival of the horse in large numbers before guns resulted in the Comanche using armoured cavalry - armoured soldiers on armoured horses, with the lance as their main weapon. Their armour was rawhide, well able to resist arrows. The heyday of Comanche armoured cavalry was the first half of the 19th century; as guns became more common, the value of such heavy armour fell. The Comanche appear to have captured some Spanish armour, with one of their well-known warriors and leader nicknamed "Iron Jacket" for his use of Spanish armour (he was born c. 1790 and killed in battle in 1858). He had inherited his armour from his father (or uncle) who was known to the Mexicans as "Coat of Mail".
Southern Mesoamerica was less suited for large-scale use of horses, and it is no surprise to see less use of them in the protracted conquest of the Yucatan. Spanish arms and armour would have been used in small numbers, again limited by the lack of native manufacture. Further south, some peoples, notably the Mapuche people south of the Inca Empire, used guns in their resistance to the Spanish. For some discussion of this, see the reply by u/CommodoreCoCo in:
References
For the use of horses by the Chichimeca and others, see
Forbes, J. (1959), "The Appearance of the Mounted Indian in Northern Mexico and the Southwest, to 1680", Southwestern Journal of Anthropology 15(2), 189-212. https://www.jstor.org/stable/3628807
Powell, P. Wayne. (1952), Soldiers, Indians & silver: the northward advance of New Spain, 1550-1600, University of California Press.
A map showing the Chichimeca nations (in central Mexico): https://commons.wikimedia.org/wiki/File:ChichimecNations.png
