Felipe Anderson is sidelined for a month, while Lukasz Fabianski, Ryan Fredericks, Jack Wilshere and Andriy Yarmolenko are also out.
It means West Ham are down to the bare bones for a fourth-round clash against the Championship leaders.
Moyes is taking a cautious approach with Michail Antonio following his return from his latest hamstring injury, and is on the bench again after coming on as a half-time substitute against Leicester on Wednesday.
Moyes has to manage him carefully with a game against Liverpool to come in the Premier League next week, so there is an opportunity against West Brom for £8million summer signing Albian Ajeti.
The Swiss forward has yet to score since joining from FC Basel.
West Ham lineup: Randolph, Zabaleta, Balbuena, Diop, Cresswell, Sanchez, Rice, Fornals, Lanzini, Ajeti, Haller
Subs: Martin, Ogbonna, Cardoso, Noble, Kemp, Ngakia, Anontio