No. ShipStation is a SaaS (Software as a Service) app. You log in to ShipStation using a browser (Chrome, Firefox, Safari, or Microsoft Edge), so no download is necessary.
To print directly from ShipStation to your label printers, without downloading or using your browser PDF functions, you will need to download ShipStation Connect. This is a small app that acts as a bridge between your ShipStation account and your printer workstation.