If you want to run Ethernet over your installed Coax you can use MoCA adapters (stands for Multimedia over Coax Alliance). It will give you a hardwired Ethernet port out their, then from there you could add an Access Point to broadcast your WiFi or even add a switch for additional network ports. They are reliable & really don't have any complicated setup... We have used them multiple times in the past on client projects when preexisting coax is in place & running new Cat 5 or 6 is cost prohibitive.
Here is one company:
MoCA Network Adapters - Actiontec.com
https://www.amazon.com/Actiontec-Bo...40-20&linkId=0d0a5acbedbcfa78e917ebafe99eda5d
Just check the Manufacturers Spec for maximum Coax length (the MoCA's spec is 300')... but I know some manufactures state greater & I know we have at least (3) unit in the field running 350+ (at a golf course for their out buildings)
This looks like it might be the ticket! As I prefer not to open another trench up to the shop.
Thanks for your help!