"World without trade" is a major change from the existing situation in the world. If it was possible to make this change, its impact will be disastrous not only for America but for the entire world. In such major changes it is difficult to predict or estimate the prices of the product of one company like Logitech in one market like America. For all that we know, Logitech may be one of the thousands (or even tens of thousands) of the companies that will be forced to close down, if world trade was to close down completely.
World trade refers to buying and selling across national borders. World trade adds to efficiency and productivity of all the countries through the mechanism of division of labour working at most macro levels. It enable a country to produce and export what it can manufacture most economically, and in return import other things from countries that find it advantageous to export them. In this way, each country is able to get a total package of goods and services that is better than what it will be able to produce by itself.
A world without trade will make every country in the world a poorer country, and reduce the average income and prosperity of people in every country. This means that the total production as well as demand of every type of product, including Logitech products, will decline. The actual price of product will also depend on other factors like rate of inflation. But the price of most of the products in real terms will definitely tend to rise because of lower efficiency and higher costs.