If I have a bitcoin wallet on say Exodus, but something happens to the people that make Exodus, could I upload the 12 word private key on a different bitcoin hot wallet such as electrum and get my coins back?
The easiest way is to create a new wallet and send the full balance to that new wallet. It is better because RE-use a private key/wallet-seed from a closed-source wallet is a security issue.
about your question, Yes, seed phrase generated by Exodus is in BIP-39 format
[1] which can be imported into any wallet support it (for electrum just use BIP option)

[1]
https://www.exodus.com/support/en/articles/8598680-how-do-i-import-my-bitcoin-wallet-into-electrum