A managed package license and a customer community license are two distinct licenses. The former is a license that's issued by a Salesforce Partner (also referred to as an ISV) which has specific use terms that you'll want to refer to. The latter is a license issued directly by Salesforce. The objects from a managed package may or may not be accessible from within a community. Similarly, the vendor's application may or may not be supported for use in a Community by your Users. That's something you'd need to check with the vendor you've purchased the app from.
That having been said, typically one can write records to most any custom object from within a community, either custom objects the org has created, or those from a managed package. However, in the case of a managed package, you'd still need to check to see if your license allows for that.
In some cases, you might need to create a "mirror" object in the Community that allows you to pass the records from the Community into your Org via a trigger before they can finally be written to your managed package because of permissions on the object. It all depends on the kind of functionality that you need and what your license will allow.