If it's not too late, it's always a good idea to take a picture of the equipment that the seller is removing. It's always helpful to know what was there and how it was configured.

To answer your question, yes, you want to buy a multi-channel amp to power the speakers in your rooms. It's important to know how many rooms and speakers per room you will be powering to inform the requirements of the amp.

Another important consideration to determine is how the owner controlled the system, and how you would like to control the system? Are there volume controls or keypads in the rooms? Would you like to control the system via your smartphones?

A couple of options for you: SpeakerCraft, Nuvo Player Series, Sonos

What about offering to buy the electronics from the previous owner?

I would suggest using Sonos as the source. Either Connect:amps if power is needed or if its full Bose then you will be stuck in its eco-system, but can use Sonos as the source for all the different music streaming services. Bluetooth is OK, but has range issues 15 ft.