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A law firm's product is legal services.
The IT for the law firm wants to purchase and deploy new laptops. Current laptops are EOL. e.g. Slow, insecure, incompatible.
Is this within realm of "The Product"
It depends on the perspective.
From the perspective of the law firm, the IT department provides a service to them that enables them to provide their product / service. However, from the perspective of the IT department, their product / service is the tools and technology that enable the broader organization to function.
So, from the perspective of the IT department, the consumer is the other employees of the law firm. This could be broken down into several categories by department, role, or some other means. The buyer would be the IT department itself, since it probably has an allocated budget or needs to conform to some kind of rules. The producer is the IT department as it does the work of providing a product or service to customers. The product is the services itself, or in this case the specific tools and technologies provided.
It looks like some of your confusion stems from terminology. What you seem to be doing is developing a use case or business justification, rather than defining a project or product. From a project management perspective, a product is a deliverable or outcome generated by a project. In other words, a project produces something; that's the product!
You appear to be conflating "product" in the project management sense with the goods and services (confusingly also "products") a company sells to its customers. A project (especially an internal one) need not be directly related to the products or services the enterprise provides its customer base. The product of a project can be tangential to the core business, although it should still theoretically add value to the business or the organization in order to justify allocating resources towards it.
Again looking at this from a project management perspective, a hardware purchase may be either a project or an aspect of service delivery. In IT service models like ITIL, a hardware refresh like this might be a cyclical part of the ongoing asset and configuration management process. The business might also consider consider hardware refreshes or capital acquisitions as part of a continuous lifecycle, and thus more of a process than a project.
Alternatively, PMI defines a project as follows:
A project is temporary in that it has a defined beginning and end in time, and therefore defined scope and resources...[a]nd a project is unique in that it is not a routine operation, but a specific set of operations designed to accomplish a singular goal.
A routine update of computer hardware is probably part of ongoing IT operations, while a large-scale capital acquisition and company-wide rollout could be considered a project. As a purely pragmatic matter, how it's categorized has more to do with how the business chooses to structure the activity (and, by extension, how the activity is funded and performed) than anything else.
As the executive sponsor for the activity whether it should be structured as a project or as part of an ongoing service or business process. At your company, that's likely to be the only answer that actually matters.
