Technically quite possible, Practically not at all.
It would fail for the same reason that such systems failed to gain popularity in to open public for us.
Intrinsicaly, the pneumatic tube system is one-way. You load the tube, launch it, and ..... That's it.
It does not include any means of verifying that the delivery succeeded.
It does not include any means of verifying that the correct person received the delivery.
It (usually) does not even include any means of verifying that the exit point of the tube is open, available and unblocked.
All of these shortcomings can be worked around using fancy mechanisms, but that requires fancy mechanisms. Which impose their own cost, failure points, and complexity to the system.
Damage of delicates:
The delivery system is inherently violent, with lots of sharp accelerations and stops. And people will inevitably try to ship stuff that is not strong enough for the system.
(anecdote: Back in the '70s ,at a local store that used a pneumatic tube system to deliver invoices, money, and office stationery to and from the cashiers:
Cashier sends a message "my pen is dying, send a replacement please".
Instead of a ballpoint pen, the warehouse sends through a fountain pen. Which shatters upon arrival, splashing cashier and several customers including myself with black ink.)
An air leak, or a misaligned cartridge seal, or something sticky either on the cartridge or in the pipes, brings the whole system to a screeching halt. Then someone has to get the rod, and start reaming the system to remove the blockage, then again with a rag to clean it, then again with another rag to either oil or powder the tube, depending on the system. BIG downtime.
People are idiots:
The pneumatic tube system is very much open to abuse, and inevitably some braindead person will play with it. "What happens if I put a can of Cola in this thing, the size seems about right", they wonder.
Or, "Hey Pete, you want a HotDog? Exxxxxxpress delivery coming right up!".
These abuses inevitably lead to the scenario above, where the Man with the Rod needs to be called out.
Pneumatic tube delivery systems work fine, in a professional environment with trained, disciplined users. For frequent delivery over relatively short distances between a very small number of recipients.
Over long distances, or where the number of recipients become many, or the operators are mere Humans.... They are not at all optimal.