There is no "the Segwit chain". That chain is Bitcoin, and all Bitcoin wallets support it and use it regardless of whether they have implemented segwit itself or not.
Will I run into any issues transferring BTC from, say, Coinbase (Segwit) to Exodus (non-Segwit) to Ledger (Segwit)? Will the middle link (Exodus which is non-Segwit) mess up the transaction in any way?
There is no problem because it is all still Bitcoin and on the Bitcoin chain.