Once a transaction is on the blockchain, it is confirmed. That is the definition of “confirmed” in the bitcoin system. So a transaction cannot “sit on the blockchain” unless it is already confirmed.
You can broadcast your transaction to the peer nodes that you are connected to, and they can relay that transaction to the peer nodes that they are connected to, and so on until pretty much all nodes on the network have heard about your transaction. The nodes will store your transaction in their memory pool in case any peer requests it. It won’t be “on the blockchain” yet, but some websites might show you the transaction if they have heard about it from any of the nodes they are connected to.