RStudio Connect is a web server, and therefore needs to be running on a machine that is permanently on, as well as permanently connected to the internet via a publicly accessible endpoint.
For this reason you have to be running Connect in a data center or in the cloud. (Although the software will run on a laptop, this scenario doesn't make sense for the type of deployment you have in mind.)
For the use case you describe (small number of users, single app, limited IT skill) you should consider getting started with ShinyApps.io. This will give you many of the benefits of Connect, especially the hosted Shiny apps, with zero installation and configuration.
Some of your other options include getting started with the RStudio Connect marketplace images at AWS or Azure. (Both these page include pricing options, but my personal opinion is that this is a convenient short-term option, but could well be rather expensive to run in the longer term.)
Your final option is to install, configure and manage Connect on a dedicated virtual machine (VM) in the cloud. A VM is a virtual server, running an operating system of your choice, that you then manage yourself, or with the help of your system administrator. The cost is going to be mainly dependent on the size of VM you provision (in terms of CPU count and RAM), and you can get started with estimating costs at AWS or Azure.
It's hard to say what size of VM you should provision, since it depends to a large extent what your shiny app does (how much RAM it consumes, how much CPU it consumes). But I would usually suggest trying an instance of size m4.xlarge with 4 CPUs and 16GB RAM - this costs ~$0.20 per hour, i.e. ~$150 per month (See https://aws.amazon.com/ec2/pricing/on-demand/). If your app is really small and efficient, you can try a smaller machine at reduced cost.
I hope this helps.
Andrie