A VPN will hide your true IP from the bitcoin node you connect to, but your VPN provider will still be able to see all your traffic. If you want something approaching anonymity then you would be better off using Electrum over Tor.

Further, although a VPN or Tor would stop the node linking those addresses to your IP, it does not prevent them linking those addresses to each other. When you load up Electrum, it will essentially send a request to the node it connects to asking for the up to date balance of all the addresses in your wallet. Even if the IP that request comes from is a Tor exit node, the bitcoin node will be able to tell the same IP requested information about all those addresses, and therefore deduce they are all part of the same wallet.