Or is there an online service that has this information available?
I'm not sure if Electrum stores the time it was broadcast... the closest I can think of would be the "received time" shown for a transaction on blockchain.com:
But that is only indicative of the time that the transaction was (last) received by
blockchain.com... it may have been dropped/rebroadcast etc...
Bitcoin core seems to store the time it first saw the transaction... but if the node isn't running, that "received time" may not be when it was first broadcast etc.
Why is the "broadcast" time important to you? What are you attempting to determine exactly?