I am considering transfering my bitcoins from coinbase to electrum wallet. My question is, what happens if Electrum suddenly shuts down and their servers go offline? Will I be able to access my coins using another wallet or will they be lost?
Electrum is not an ordinary wallet that will shut down their servers suddenly. They serve their users better 24/7 than other wallers.
In whatever way, you need to have access to your electrum wallet sensitive credentials like pass phrase, private key.
You will be able to access your bitcoins using other wallet that supports import of private key in case you face issues with electrum.