There are several different types of FM systems available. Depending on the situation one type may be better than another. You may want to follow up with the audiologist about what type he/she recommends.
A classroom or Sound field system is a speaker that increases the volume slightly for the whole class it is typically positioned at the back of the room so that the student is receiving input from both the teacher and the speaker. This is a permanent speaker that stays in 1 classroom.
Personal FM systems are usually a small speaker sitting on the students desk which puts the teacher's voice directly in front of the student. The speaker is small enough that the student could bring it with them to another classroom if needed. About the size of a lunchbox.
People who wear hearing aids usually have a "boot" or other receiver that hooks to the hearing aid. They would attach this to the aid when they needed their fm system on. This could move with the student from class to class as long as the teacher had a mic.
I had a student in high school who did not want anything that was noticeable to other students. He was deaf in one ear, so he wore it on his hearing ear. He wore a small receiver that looked like a hearing aid. It was very small and sleek, here is a link:
Something like that though would only be appropriate for an older student who was able to take care of it and put it in unassisted. It is also portable enough for the student to take it between classes.
All of these systems require some type of microphone that the teacher wears. I found that teachers prefer the lapel or necklace type mics over the boom microphones (the headband type).
If this is something required for your son in his IEP then the school has to pay for it. The school would own it and your son would use it as long as it is written in his IEP. If you want something for home as well as school you would need to purchase it.
I worked in the school system as an SLP and was responsible for putting them in for students who needed them, I consulted with an audiologist about placement etc.