The supporters group, chaired by Wallace, officially took ownership of the Tynecastle club earlier this week but Hearts will continue to be run by an executive board led by chairwoman Ann Budge and chief executive Andrew McKinlay.
The Foundation have previously come under pressure from fans to make public their views over contentious issues at the club, particularly when former manager Craig Levein and current boss Robbie Neilson were coming under heavy fire from fans.
Wallace vowed he will never shy away from tough discussions but insisted they will always be held in a private and professional manner so as not to undermine the club's dignity.
He said: "The one thing we need to be clear on is that when we're told by fans that we need to challenge the board on certain issues, we need to remember that two of us are also members of the board.
"We will challenge things behind closed doors. I've always said I would never challenge the club in public.
"That model where members of the fans' group have come out and talked openly about board issues has been trialled elsewhere and failed. We sit in the boardroom and have challenging discussions.
"We have great discussions and we have difficult discussions. It should always happen within the confines of the boardroom. I can assure everyone involved with the Foundation of Hearts that we will always raise supporter concerns in the boardroom. Ann has always welcomed that type of challenge for as long as I've been involved.
"Often the club board are well aware of the vibe within the fans but I do think it's important that we at the Foundation keep our finger on the pulse. Now and again Ann will ask me what the view is in the support on certain issues. We will always listen to the fans and engage with them but it's important that it's fed back to the club in a constructive way."
Wallace paid tribute to two of the club's former players for their efforts in helping Hearts fans take ownership of the club. Donald Ford, who played for the club in the 1960s and '70s, brought his accountancy skills to the group while Gary Mackay, the club's record appearance holder, was heavily involved in bringing key people to the table, including Budge.
Wallace said: "Donald and Gary were crucial. Donald was my first ever hero at Hearts and I got to meet him through this. He was one of the original five behind the Foundation of Hearts and he was pivotal.
"Gary also played a huge role. He knew of Ann Budge and must take credit for bringing her to the first meeting with the supporters' group and persuading her that it was a conversation worth having.
"Gary and I are the same age and I spent my formative years as a teenager into my 20s following that great team in the 1980s that Gary was such a big part of. As the club's record appearance holder, it's fitting that he was able to play such a big part in bringing Ann to the table because without that, this (fan ownership) wouldn't have happened."