PCI DSS 3.2 states: "PCI DSS applies to all entities involved in payment card processing—including merchants, processors, acquirers, issuers, and service providers. PCI DSS also applies to all other entities that store, process, or transmit cardholder data and/or sensitive authentication data."
There is an FAQ on the PCI SSC site that asks if PCI applies to paper with cardholder data and the answer was: "Yes, PCI DSS requirements are applicable if a Primary Account Number (PAN) is stored, processed, or transmitted by any media, including paper records. PCI DSS Requirements 9.5 through 9.8 specifically addresses the safeguarding of physical media, including paper records, containing cardholder data."
Given those two together, it looks like it would apply in the case of handling the media of the plastic card. As Jeff pointed out in a comment, if you don't have an agreement that contractually obligates you to PCI DSS, applicability gets interesting.
Not knowing the particulars of the process you're asking about, what I'd look at is whoever is sending the cards to the company in question. Are they a merchant or service provider that is obligated to be PCI DSS compliant? That company should be looking at 12.8, "Maintain and implement policies and procedures to manage service providers with whom cardholder data is shared, or that could affect the security of cardholder data, as follows:". This is followed by subrequirements that talk about written agreements that acknowledge responsibility, diligence processes to understand who has what responsibility on different requirements, etc.
In my experience with QSA audits, what I would expect to happen is the QSA would request a list from the merchant that included all entities with whom the merchant shared card data in any capacity. Then they would dig into that. While your company might not have any direct obligations, the merchants/service providers that you're working with that do have obligations could be in a bad situation if they needed to show evidence that they knew you were PCI DSS compliant and could not.