When you have the dash control unit in the vent on over to the air conditioner control selections and you are pressing on the accelerator, the air will quit blowing through the vents. When you let off the gas the air will blow again. It is a vacuum controlled unit. I have tried other dash controls and it still does the same thing.
Common problem, simple fix. Vacuum goes down when you accelerate, and for safety, these systems are spring-loaded to go to defrost mode when vacuum is lost. That means your feet might be cold but your windshield will be clear.
Look for a storage canister or check valve inline with the vacuum hose going through the firewall. Some cars use just a check valve. There is a larger valve with a small storage area to correct the problem you described if your vehicle only has a check valve now.