The 23-year-old Scotland international right-back has joined on an intial one month loan deal and should make his debut against Cardiff this weekend.
Hammers managing director Paul Aldridge told the club's official website, www.whufc.com: "We are pleased to welcome Robbie to the club and we trust he will settle in quickly."
Meanwhile the on-loan transfer of Crystal Palace utility player Hayden Mullins has now been turned into a permanent deal.
Mullins, who made his Hammers debut in the 1-1 draw with Nottingham Forest on Wednesday night, has signed for an undisclosed fee.