Yes. You could build a cluster of Shiny Servers and load balance traffic to them with a proxy like nginx. The key thing is to make sure to have sticky load balancing so that future requests for a given session go back to the same server. Please see this article on running Shiny Server with a proxy for related information.
Note that that article is much more focused on nginx as a proxy than on nginx as a load balancer, but fortunately, configuring nginx as a load balancer is no different for Shiny Server than it is for any other web server.
There is an article on setting up nginx load balancing here. To load balance Shiny Server instances, you can put the addresses to your two Shiny Servers in the "upstream" property in the nginx configuration file, using the ip-hash load balancing mechanism. This mechanism is necessary to ensure "sticky" client sessions; otherwise, the applications will break in unexpected ways. Finally, configure the "map" and "server" components of the nginx configuration file as described in the "Running Shiny Server with a Proxy" article, and restart nginx.