Yes, MSSPs can use Threat Response. In fact, there are more than 100 MSSPs actively using Threat Response in their investigations. Threat Response partially supports multi-tenancy, in that you can configure as many copies of a module as you need, one per customer. Currently, there is no true multi-org support, but you could build your own by creating accounts to each organization/customer as a distinct Threat Response organization and then using the APIs to reach and manage each environment.
Learn more about Threat Response here, or check out other FAQs here.