USB-C Power Delivery with a buck converter
Yes, the standard way in PD negotiations (as in many other cases as QuickCharge, or even basic USB device functionality) is to start with low-power electronics and conduct power negotiations. And then turn on power function if negotiation gets successful. For example, USB must have most protocol capability when running out of 100 mA source. So this is not a problem, and this is a commonly suggested solution.

When doing PD negotiations, your high-power "function" (5V@6A) must be disabled, most (if not all) of buck controller ICs do have some "enable" pin. Your PD logic must keep the power function disabled until successful negotiation into high-power mode. More, the buck controller ICs should have so-called "undervoltage" protection, so the explicit "disable" might be unnecessary. 

This means that you might need two microcontrollers to accomplish your high-power function, one for PD sequencing/control, and another one that needs 6A to run.