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?
The servers are managed by multiple people, even you can create one so having them all shutdown is not possible. They also don't store any sensitive information's so no worries, continue to use it. If things goes wrong (unlikely), export the private keys or import your seed to another BIP39 compatible wallet.