The 28-year old goalkeeper joined the Hammers from Birmingham City on the weekend and was thrilled to be given the opportunity at Premier League level.
"My initial reaction to joining West Ham is excitement," the Irishman told the club's official website.
"It was a tough decision turning down Birmingham's offer of a new contract because of the second part of the season we had and the team we had built.
"They are a great group of lads and staff but I felt like I couldn't turn down the chance of joining a club like West Ham. Everybody wants to try and get to the top of their profession and I'm no different."
Former Motherwell number one Randolph has departed St Andrew's after two years with the Birmingham club.