Benrahma was sent off before the hour-mark as the Hammers crashed out of the FA Cup following a 1-0 replay defeat against the Championship side at Ashton Gate.
Moyes was without Jarrod Bowen, Michail Antonio, Lucas Paqueta and Pablo Fornals for the trip because of injury, while Mohammed Kudus is away at the Africa Cup of Nations.
There is hope that Bowen and Fornals could return for Sunday’s Premier League meeting with Sheffield United, but Benrahma - who was the subject of rejected loan bid from Marseille on Tuesday - will now be suspended for three matches.
Asked whether he anticipated any new signings before the end of the window, Moyes replied: “I would probably say yes to that.
"The truth is we’ll try and do some business. Undoubtedly, there’s a desire from the club to try and do something and from me and everybody there. The next part is, can we get the right players to come in and ones we think could help just now?”
Moyes had no complaints over Benrahma’s dismissal, which came after the Algerian kicked out at Joe Williams in retaliation for the midfielder’s late tackle.
Both Williams and teammate Taylor Gardner-Hickman were, however, fortunate to escape red cards over separate incidents, with VAR only in use at Premier League grounds during the competition.
"It’s incredible that they call it the best cup competition in the world but yet we have one week where we have VAR and the next we don’t,” Moyes said. “I find it quite ridiculous. It’s not level for every club. If we’re not having VAR we shouldn’t have it in the tournament at all. If we’re having it, have it everywhere.”