The duo also confirmed in the letter that they are close to completing their first signing of the January transfer window.
Allardyce is the only manager of a side currently in the bottom five of the Barclays Premier League who was in charge at the start of the season and, following this weekend's 5-0 FA Cup thrashing at the hands of Nottingham Forest, there has been speculation the 59-year-old may be the next boss out of work.
The one bright area for West Ham has been the Capital One Cup, in which they beat rivals Tottenham in the last round to set up a semi-final meeting with Manchester City, the first leg of which takes place at the Etihad Stadium on Wednesday evening.
And Sullivan and Gold have backed Allardyce to move the club up from 19th place in the table and, with the support of vice-chairman Karren Brady, they will also use their own experiences to overcome West Ham's current plight.
"We know Sam has not lost his ambition or desire and is committed to making West Ham United a great Premier League club," the open letter read.
"Sam has been handed a near impossible task of coming through three crucial games in the space of six days with a squad of only 14 fit and available senior players.
"We have experienced difficult times before and we know what will get us results - hard work, determination and perseverance.
"To that end, we are all working round the clock to do everything in our powers to help arrest our slump in form."
It is the seemingly ever-growing injury list at West Ham that has been highlighted as the main reason for the club's poor run, which has seen them fail to win a league game since November 30.
Andy Carroll has yet to feature this season while James Tomkins, Winston Reid, James Collins, Mark Noble and Ricardo Vaz Te are also injured and captain Kevin Nolan is currently banned.
Sullivan and Gold revealed in their letter that they are close to providing Allardyce with more players and believe once their current crop of crocked talent start returning to action things will get easier for the Hammers.
"Our first priority is to get our key players fit," the letter continued.
"Our record signing Andy Carroll continues to make good progress following his return to first-team training and he should be back on the pitch very soon...Trust us, Andy Carroll is desperate to play and is ready to do his bit for the team and we are confident that the return of all of our injured players will lead to an upturn in our fortunes.
"Thereafter, our second priority is to bring in the new talent needed to bolster our squad in the most needed areas.
"January is not an easy time to do business but we are pleased to confirm we are close to securing one of our key striking targets. A defender is high on the list of priorities as well and progress is being made on that front."
Everton defender Johnny Heitinga rejected a move to Upton Park last week and Allardyce has conceded attracting players to compete in a relegation battle may prove 'impossible' but, with talent such as Monaco forward Lacina Traore reportedly closing in on a loan move, any business will be welcomed by everyone involved with West Ham.