Bottom line is: as one of the (or the only?) Business Analyst(s) for your company, you are responsible for reviewing your business process, technical, resource, and product needs. So if you have requirements that overlap both in-house and off-the-shelf systems, you still have to capture those requirements.
In many organizations, a systems engineering or administration group would then be responsible for deciding if your requirement is satisfied by an in-house application or an off-the-shelf application. A natural conversation would involve you in those discussions with the systems people and you could (should!) modify your original requirements to clarify any specific points.
If there is another individual who is responsible for the off-the-shelf applications and requirements, then you should still write your own requirement but coordinate with that other person.
Generally speaking, you are writing requirements for the overall "business". It's up to the systems admins or engineers to decide how to satisfy that requirement.
The other half of your question refers to the BA in the company making the off-the-shelf (OTS) product. Don't worry about their requirement. You only need to make the requirement clear from your company's point of view. The OTS company will accept your requirement as a customer input, then will create their own internal derived requirements from that (and hopefully modify their OTS product to support your need).