Normal method to get things into space as a private individual is to build a cubesat, comply with the rules of whoever is going to fly it, send it to them and wait till they have a slot. Some costs here but depends heavily if someone chooses to sponser the launch in some way because they are interested in the results (noting sponsor can mean things like getting assistance with engineering and testing or access to lower cost launches, not just $).
Note that those rules often restrict things like pressurised gases so reading the fine print and doing the engineering to prove you are not going to blow up an entire launch on the pad is critical to avoid being rejected without a refund.
If you just want to get your payload up high a sounding rocket will do it for lower costs but skips true re-entry.
If you have a concept for a space experiment you can also work through an educational organization that is building cubesats as part of engineering and propose your concept to them, which potentially gets it to fly for free but requires a concept both interesting and practical for their budget and experience.
A more pragmatic problem for your specific question is 'how are you going to tell what balloon did what'. By themselves they will re-enter randomly, if you build a cubesat with a RCS system to de-orbit exactly over your home, all the action will take place 100km up at a couple of km a second. Even if your cube sat has sensors on it, it will be flying a very different trajectory.
A more practical way to actually investigate the balloons from orbit question is to simulate it, either yourself or by paying to have it done. If results are interesting do runs in a high altitude wind tunnel and only then actually attempt to fly something in space, knowing a fair bit about what could go wrong and waste the whole thing.