Well, the only required rooms in a vault are Living Quarters, Power Generation, Diners, and Water Treatment rooms in addition to the Vault door and Elevators to connect the levels. The other room types are not "needed".
If by accommodate you mean having a room assignment for each dweller, you will need 100 rooms total. This number includes the Vault Door but not the Elevators. If you don't mind some of your dwellers being on permanent coffee break and wandering around, you will not need that many rooms.
At a minimum you are going to need five triple Living Quarters fully upgraded to take in 200 dwellers. That is a total of 15 rooms.
After that, the numbers get kind of fuzzy, because production over time is based on the SPECIAL attributes of the people assigned to the rooms. If you wish to maximize production you will need dwellers with 10 in the appropriate SPECIAL attribute for each room. Assuming you are limiting yourself to the basics, that means at a minimum you will need one Weight Lifting room, one Armory and one Athletics room to train dwellers for power, food and water production.
You also have to figure that incidents cut into production. That is to say, if a fire breaks out in a Power Generation room, that room will not be producing anything until the fire goes out. If raiders attack, they will start stealing your resources (and of course, production will stop in any room that they are in) until they are killed or driven off. As a result, you will need to produce slightly more power, food and water than you can actually consume.
I have never seen a fully optimized vault running a minimalist configuration with 200 residents, but as a guess, I think it could be done with around 50 to 55 production/residence rooms, but it will be a slow build. That is just a WAG, so take my guestimate with a grain of salt.