I don't know how to contribute to our team's monthly planning. How should I come up with ideas for things that the team should work on and contribute to our team's road-mapping meaningfully?
There are 2 ways to contribute in these meetings:
As you work, keep a doc of potential improvements/ideas that you wish you could work on, given more time.
Before the roadmapping meeting, meet with your eng manager, senior engineers, and product managers to better understand their priorities. Ideally it should not be a surprise to you (or anyone) what the priorities are in the monthly plan.
Since you have a rough idea of what will be discussed, spend some time going deep in one area. Become the most knowledgeable person about the risks, how long it'll take, and which teams need to get involved. At a company like Meta, there will almost certainly be other teams involved -- perhaps you could talk to them.
See these videos for building relationships:
You want a new job but you have no experience doing this job. If the job is vacant and they are desperate enough they will be open to letting someone without much experience do it. That's one way to go. If your team doesn't have any E5 or PM, or maybe for some reason nobody takes care of the road mapping, nobody will question you if you step up and do the job. At least in Meta. Nobody will challenge your qualification if they don't take that job themselves. It's not uncommon to see an E5 to E6 promotion happen in this way (when the team has an E6 job without an E6).
This is no longer commonly available for E4 to E5 nowadays. Sometimes a team has so many capable E5s that they are competing to do road mapping among themselves. You won't be able to take their job out of their hands. What you can do is add value to what they are already doing. You take a small piece of the job you want but you don't claim you are taking it. You are just "helping out" by sharing some ideas and giving some suggestions. If they are not good enough, it's not a big deal. Nobody thinks you are stealing their jobs.
Take your time practicing the job without officially taking the job. It's like shadowing someone else doing that job -- asking questions and sharing your thoughts. When you are experienced with the job it's easier to convince others to let you take the job. If you can do it better than the existing owner people will notice. They will ask you to do it instead of the existing owner.