Will using a 10/100 router stunt gigabit switch speeds for an internal network?

I have a 10/100 Verizon router. 3 computers are connected to this via Ethernet. I'd like to put a gigabit switch between the computers and router to enable gigabit speeds on the intranet.

If the machines are still using the 10/100 router as a gateway will this stunt the intranet speed? Or will they actually communicate with each other at gigabit speeds?

Yes, the router will only allow the machines connected to it to function at 100 Mbps. So, no, they will not be able to communicate at gigabit speeds when connected only to the router.

However, since you have a gigabit switch. Connect all the machines to the switch and then run the switch uplink to the router. The port used as the uplink will connect to the router at 100 Mbps, and the rest of the ports on the switch will function at 1000 Mbps. I have this exact setup. All the machines can then communicate at gigabit speeds on the switch and your 10/100 router can still be used for internet/wireless/etc.

I must point out that there are several folks on here who have put gigabit switches in front of their 100 mbit router, and are seeing performance degradation. A simple reason for this is that the hosts connected to the switch are sending it data at line speeds the switch cannot offload to the router fast enough for buffers to not overrun, and drop frames.