You need to always use fully-qualified domain names when talking to vCenter. If you don't have DNS, you need to get some because, without it, vSphere will not work properly.
The vCenter have DNS ,"VMDNS" server.
The domain "school.local". It work fine.
But it only for the vCenter PCs.
The remote-PC use the "DNS2" server for outside internet. It not config for domain school.local.
Does any way to solve remote-PC connect via vSphere Web Client problem with no change the "DNS2" server?
vmdnsd service no longer used in 6.x onwards
sorry for my topology mistake.
The "VMDNS" is a windows server DNS service use for this vCenter.
not a vmdnsd service.
The PC in the other subnet needs to have a forwarded available to it in the DNS server where it's pointed. All clients must be able to resolve the hostname of the vCenter Server, so if the first DNS server cannot within school.local then it needs to forward to another which is authoritative for that zone.