The League One strugglers knocked Sunderland out in the fifth round and have been rewarded with a quarter-final tie at Chelsea.
Liam Rosenior’s side will start as big favourites to go through.
Should they reach the semi-finals, their Premier League trip to the Amex will have to be re-arranged.
Chelsea are currently due to visit the Seagulls on Sunday. April 26, which is semi-final weekend.
That will have to be changed unless Vale pull off a huge upset at Stamford Bridge in the sixth round.
Options for a midweek date will become clearer when it is known whether Chelsea have gone past PSG in the Champions League.
Premier League leaders Arsenal will head to Championship side Southampton in the last eight while Manchester City will be at home to Liverpool.
Leeds, who comfortably beat Norwich 3-0 on Sunday with Facundo Buonanotte on the bench, will travel to West Ham.
The Irons edged past Brentford in a penalty shootout at the London Stadium on Monday night, the tie having finished 2-2 after extra-time.
The quarter-finals will be played around the weekend of April 4.
Brentford boss Keith Andrews criticised the “disgusting” culture of ridiculing players who miss penalties after Dango Ouattara’s attempted Panenka back-fired.
His effort was easily caught by West Ham goalkeeper Alphonse Areola as the Bees slipped to a 5-3 shootout loss at London Stadium.
Brentford boss Andrews alluded to the host of England internationals who have been “persecuted” for failing from 12 yards at major tournaments during a passionate defence of the Burkina Faso forward.
Hammers captain Jarrod Bowen and Bees top scorer Igor Thiago each scored twice during normal time as an entertaining fifth-round tie ended in a 2-2 draw after 120 minutes.
“No, I’m not annoyed at all,” Andrews said of Ouattara’s penalty.
“I think the easiest thing for a footballer to do is not take a penalty. It takes unbelievable courage on a stage like that to take a penalty.
“It’s probably the first time I’ve ever spoken about penalty kicks and people that miss them.
“I despise the culture around players that have missed penalty kicks – I think you know the ones I’m talking about, national heroes that have done it, ridiculed, persecuted. I think it’s disgusting.”