Normally, the project manager obtains permits and inspections.

If you hire a general contractor or if your architect is managing the project, s/he will obtain the permits. If you are managing the project, you will need to do this. If you want to get the permits, you can discuss this with your contractor.

The most important thing is to be sure you and your contractor are clear about obtaining permits so you don’t start work and then realize you don’t have the necessary permits.