Top scorer Stockley broke his nose during the 2-1 Sky Bet League Two defeat at Newport on New Year’s Day.
Skipper Moore-Taylor missed that game with a muscle injury and Grecians boss Paul Tisdale is already short of defensive cover.
Troy Archibald-Henville is nursing a knee problem, while fellow central defenders Troy Brown and Luke Croll have been absent for several weeks.
Matt Phillips and Craig Dawson are out for West Brom.
Winger Phillips (hamstring) and defender Dawson (fatigue) will not travel for the third-round game at St James Park.
Nacer Chadli (hip) and James Morrison (Achilles) are also sidelined but Sam Field, Oliver Burke, Allan Nyom and Boaz Myhill could all feature. Jonathan Leko may also travel after returning from his loan spell at Bristol City along with Kane Wilson, who has spent the last five months on loan at Exeter.
Northampton striker Kevin van Veen is hoping to make his first start in over a month in the home game against Shrewsbury.
Derby's decision to postpone Sunday's home Sky Bet Championship game against Cardiff was made independently, Derbyshire Police have confirmed.
Rochdale have no new injury worries ahead of Tuesday's Sky Bet League One relegation battle with fellow strugglers Fleetwood.