How to ask EM to use effective sprint planning?

While sprints and agile frameworks are commonly used in software industry, not every team needs it. Can you provide more detail on how you your team's productivity/execution/teamwork is impacted by the lack of one? What do your teammates think? If your team / company is small, full agile-based sprints might be more pain than it's worth. Knowing these details might help you convince your teammates or your manager if it's necessary.

I want you to know that I'm concerned about communicating this request for proper planning with my manager.

Hmm, this is a bit strange. What concerns you? It seems like you are looking out for the team as a whole, and you don't come off like you are here to criticize your manager. If you mention this as a personal curiosity like "I've been wondering why we don't have sprints/agile process" or an suggestion you want input on "I have proposal that might help the team and wanted your opinion on it", any reasonable manager would be happy to talk about it.