SImple question - wifi and Hardwired connections

You have to move away from thinking about Wifi or Ethernet and think more internal network (LAN) and WAN (Internet). The computers use a server, so first you build your LAN (local network) around that, then you stretch that 'connecting point' (normally a switch or Router) to a external source (Internet Provider - ISP), and you design it in this methodology to ensure both optimal settings, and high security of your systems (do you want someone to just remotely fill your server with child porn then call the cops to watch them 'bust you' for the LOLZ?).

First why are we changing from Wifi to Ethernet, there needs to be a justification for the change and potential hiccups that can result, or "unseen" results of the decision?

If the concern is 'how slow the Internet is" that Ethernet will speed things up, this is a BIG mistake, and incorrect. Internet connection speed is what you paid for, so if you buy into Cable Internet service, the MAXimum speed (like a Highway speed limit) is 30Mbps, some have been updated to 50Mbps and the goals for most is 100Mbps. But this means from really slow 100Kbps UP TO 100Mbps depending on many things from your provider, so changing to wiring wouldn't resolve anything if that was the issue.

If the concern is how slow it is to get to the server, this can again be different reasons (including the server is too low end to support the demands being pushed on it at the moment, etc.). Honestly 802.11AC (the current standard) has replaced the older 802.11n and very old 802.11g/b Wifi, and is cheap. It provides speeds wirelessly between devices (between the 4 computers and server) around 300Mbps to as much as 1Gb depending on conditions and traffic. This provides maximum portability, and is easily secure, but most of all if the 4 computers are NOT desktops then daily 'plugging and unplugging' the Ethernet normally leads to broken 'ethernet ports' on the laptops and requiring the entire laptop replaced (you can't swap out the ethernet card like a desktop).

If Ethernet is the only choice provided, though this information was provided, then as mentioned you build the 5 computing systems (4 desktops 1 server) to a switch, normally a 8 port one, so port 1 or 8 is wired to the ISP combo router / modem (unless you been provided a 8 port router). You then would need to either coordinate with the ISP (you should represent this is a business, and yes this makes it more expensive service, but you get extra services for the costs) to setup the router to maximize the services used via the server (your LAN is those ports connected together) or you would need someone familiar with the router to setup and maximize the priority (QoS - Quality of Service) that the Server gets over Internet services (Youtube, Webpages, etc.) a computer may request. All depends on what the server is used for, what are the primary daily tasks of the 4 computers and so on. More detail helps qualify the necessity of decisions to the design of your network.