Does anyone have any recommendations on how to best frame a conversation with finance to secure additional budget for sustainability initiatives? Are there any buzzwords or phrases that finance loves to hear?

Like any conversation involving budget allocation, you must demonstrate the ROI. This can be direct cost implications, waste elimination, reduced risks, improved compliance, or customer satisfaction. Make the argument using facts and ensure the messaging aligns with the organizational strategy. I think it is less about broader industry buzzwords that float around in mainstream media and more about using the vernacular of the company to support its strategic vision, business outcomes, and overarching goals. Competing priorities will also inform budget decisions; the key is to get the conversation started, elevate the importance of the initiative, and develop the business case. It may take more than one conversation, so be patient. 

Here are a few thoughts

Dollars follow carbon, so if you reduce carbon you are typically saving money.

Reducing uncertainty, unreliability and variability will reduce waste this will save you money and increase utilisation

I hope that helps