I contacted Custer Township which owns Oakwood-Braidwood. Here is their response:

The records for Braidwood Oakwood Cemetery are public records. There is no charge to search the records. If you were to request something like a hundred copies to be printed, there might be a service charge to print copies. The records are fairly good from around 1873 until around 1882. Then, there is a gap until around 1901. The records then are somewhat sketchy until around 1910, or so. Then, there are no records until around 1942.