Carb rebuilt, engine still hesitates.

I am guessing. Carb or ignition problem are both possible. If the distributor advance is stuck, the engine may hesitate due to lack of timing.

The vacuum thingy is likely the advance control that is part of the cleaner air package.

If you have a timing light, test timing advance when increasing RPM, and w/wo distributor vacuum hose. If the timing light goes out as RPM is increased, then there is an ignition problem.

If all is good, then it could be the enrichment circuit in carb. The first place I would look would be the metering rod piston, if the spring is in there. At idle the metering rods are down. If the spring is missing, or they are stuck down due to an assemby issue, engine will hesitate.

Also check the float level, and that it drops without hanging up.