So the diagram has a switch, is a Switch required for the Instant On AP to function properly?
And merchant processor/p.o.s machine is connected to the Router (Gateway) or into the switch?
A switch is not required. The only thing that is required is the AP has to be able to obtain an IP.
If the pos device is wireless then all you would need is an Instant On AP connected to your modem. If the device is wired you could use an AP11D which will support wired devices as it has ethernet ports for clients.