It is definitely possible to run a Lightning node without running your own Bitcoin node, but all that means is you are using someone else's Bitcoin node for all on-chain data, and broadcasting channel opens and closes.

All self-custody mobile Lightning wallets are some form of this setup. Phoenix, Zeus' Olympus, and Mutiny before it shut down all are Lightning nodes that reach out to a remote Bitcoin node for on-chain needs.

It is far better to run your own Bitcoin node, if you can.