I have a question regarding if a switch or router is the way to go in my situation. We have a modem through Comcast that only has one available port on it right now and we are getting a VOIP system through RingCentral. We are receiving 5 phones that support PoE, most routers I am finding either don't support PoE or they only have 4 ports on them. Can I plug the Switch into the open port on the Modem and have all of them work like they should?
Switches are where you typically find PoE especially in enterprise-grade equipment. You should have no issue extending your LAN with a switch on an available port on your cable modem, but you really should have a router/firewall function sitting between your LAN and the WAN from your modem to protect your network.
You can confirm that you are directly connecting your LAN to the modem by the presence of public IP addresses which is a good way to get hacked. A firewall provides a baseline of protection that you seem to be lacking now. Be aware that the firewall (on your router) will be providing a NAT function that doesn't always play nicely with SIP packets from your VoIP. You may need a firewall capable of handling SIP properly or a VoIP system that can support STUN.