Do you have the command
in the config? It doesn’t route unless you do. On some images this is not on by default.
Then look at the routing table
show ip route
and you should see the connected interfaces (your vlan interfaces) and it will then route between them by default if they are configured as the default gateway for devices in each vlan. you are correct that you do not need a routing protocol, as all the interfaces are on the same switch/router.