Both players described the Upton Park outfit as a great club and hope to be playing Premier League football next season, as they join Ravel Morrison in east London
Both players have arrived for an undisclosed fee and signed two-and-a-half year deals with an option for an extra year.
Maynard’s 45 league goals in just three-and-a-half years at Bristol City has seen him become a much sought after player, with Wigan reported to have been interested in the former Crewe man.
However, the Hammers secured his signature, and Maynard says he cannot wait for his career at Upton Park to begin.
"I'm ecstatic,” Maynard told the club’s official website.
“It's been a long day [transfer deadline day] but I'm delighted to put pen to paper and can't wait to get going.
“West Ham are a massive club. They've got everything geared to get to the Premier League and the Premier League is where I want to be."
Maynard believes he can bring something different to West Ham’s promotion push as well.
He continued: "I'm a quick player who likes to get in behind and thrives off chances. I'll keep plugging away all game and as I'm not happy unless I'm scoring goals."
Similarly, Vaz Te is delighted to join the club and is looking forward to working with a few familiar faces.
"I feel great,” he told the club’s official website.
“I'm very excited to work with Sam again and it's great because I know a few players here.
“I'm very thankful and excited and looking forward to it.
"I spoke to Sam on Tuesday and went to see him at his office. He told me what he's looking for. I'm very excited and I think he's very excited too. I've played with Kevin Nolan and Joey O'Brien before and I also know Neil McDonald and some of the backroom staff."
While the 25-year-old is relishing the prospect of playing for such a big club, he also took time to thank his former employees, Barnsley.
He continued: "I had a fantastic time at Barnsley. I'm very grateful for the opportunity they gave me and I hope the fans can understand why I left as West Ham is a great opportunity for me.
“I have a great respect and love for the Barnsley fans and I wish them the very best."