Should I Update My Manager First or Wait to Provide Info?

In this situation, it's generally a good practice to provide a status update to your manager first. This way, they are aware of your ongoing tasks and can prioritize their requests accordingly. You can inform them about the information they requested, explain that it may take some time to obtain it, and also mention your current workload with other people. This open communication will help your manager understand the situation and make informed decisions.