Most of the time, yes, they will let you in with a mobile order. However, some have reported they’ve been flat out turned away while others have said security guard required them to hit the here now button to put through the order. Mainly depends on the guard at the gate what they allow.
worst case scenario, go park at Disney Springs and take a resort busto the resorts you want to go to. Or if you paid for parking at a theme park, leave your car and take Disney transport. I know, its not as convenient, but you can get anywhere on Disney property with Disney transportation.