If I’m an HRBP meeting with a client for the very first time, what are some impactful, open-ended intake questions I could ask my new client to help understand where HR support might be needed in their business? I’m thinking of questions that would be more likely to elicit the more strategic needs, versus tactical. Open to questions and resources. 

You can ask some questions like what changes can you see in your business over the next 5 years , what are the big people challenges that those changes bring .

But that said in a first meeting I would not avoid the tactical , because one of the keys to being  good HRBP is to be useful that means solving some of the current issues . Trust is built quite often on shared experience in solving issues.  And trust is essential! 

If you want to know what his key worry is?  Then you may ask "What's keeping you awake at night?"  This usually give you leads to probe further.

It is important to focus on your client, so indeed ask what he/she is expecting from HR for the business and team but also for him/herself as leader. Support on processes is one thing but for the leader to know you can be a sparring partner and an objective party in case there are tough decisions to take or for him/her just to have someone to connect with as part of their own development is often appreciated. Ask what the business priorities are and come up with some ideas how HR can help, avoid the operational and process topics, explain your experience and how that can help the business.