We are installing RStudio Server Pro and Connect. Will we need to run each component in a standalone or clustered environment?
You can configure both these products to run in a cluster, since both products support built-in load balancing.
However, the architectural decision whether to set up a cluster depends on the load you expect to get on these machines. The load will depend on the number of users, the frequency of jobs and the computational complexity of these jobs.
Without knowing your expected usage behaviour, it's difficult to advise on your architecture.
I would suggest you set up a standalone configuration, then measure the performance of the machines. If you find a single server is unable to deal with the demand, then start planning for a cluster setup.
You can also review the article Configuration and sizing recommendations, or speak to your RStudio account manager for additional suggestions.
I definitely agree with Andrie. The only thing that might push you into a clustered / load-balanced situation when you do not have concern for load could be High Availability (HA), so functionality is not dependent on a single computer / server being up. Both admin guides (docs.rstudio.com) have details about the specific setup / architecture for each product.