Just 1 per country. Though the country has to qualify the weight.
Here's an article from United World Wrestling about the process.
And here's a list of who has currently qualified their country for a spot. So far the only event was the World Championships in Vegas in September, where the USA qualified in Greco-Roman at 75kg and 130kg, Freestyle at 74kg and 97kg, and Women's Freestyle at 75kg. The next opportunity to qualify is to place at the top 2 at the Pan-Ams in Dallas March 4-6. Then there are two last chance qualifiers. One in April in Mongolia (Top 3 placewinners earn a spot) and one in May in Turkey (Top 2 placewinners earn a spot). There are then a few at larges that get dispersed but those go to countries that had no wrestlers qualify, the USA would never be considered for one of them. This is a separate process from deciding who from the USA gets to fill the weight. That is done at the Olympic Trials that will be held in Iowa City in April.
Right now the goal is to become eligible for the Olympic Trials.
Here's the list of everybody who was qualified so far. Wrestlers with ISU ties on the list are Angel Escobedo at 57kg, Jon Reader at 86kg, and Jake Varner & Kyven Gadson at 97kg.