Hello. If your pc allready has 4 hdmi inputs you dont need any other capture card,if you not have you need. What is your capture card-hdmi inputs you have now? There are hdmi capture cards with 4 inputs in the market.
The most important thing that you must see also is about cameras is if they are able to be active and send signal from their hdmi port all the time without going into stand-by (that will drop hdmi output) after a amount of time. My experience with mirrorless cameras not good for continous streaming. Either they close after 30-40 minutes of use (basically standby and send hdmi signal) or if we have sd card and capture (to avoid close after 30 minutes) they stop to send hdmi signal over their port,so no use for streaming or they dont have a way to send 100% clean hdmi signal and in the hdmi out you see the controls that you have in the internal display
You must see if your cameras are able to send clean hdmi signal for all time in order to not to have suprise after

Are you sure the 4 HDMI plugs on your computer are inputs? Are you sure they aren't outputs from your graphics card? Your graphics card cannot be used to capture video from a camera. You need a separate capture card.

Within the EU there used to be an issue, where photo cameras which could also output video would stop output after 30 minutes. The reason were different import customs rates for video and photo cameras. Things have adapted since and this should not be an issue for DSLRs imported after 01/2021. Maybe older modells can fix this with a firmware update. However, some photo cams run hot and shut down after a while when used for video. You do not need the SD card while streaming, the signal goes directly out of the camera on the HDMI.

As nikosman said - make shure you have indeed a HDMI capture card and you are not taking the graphics card with 4 HDMI outputs as an input.

Good luck!