Pushbuttons dont usually need to be replaced with an identical model. In your case, I beleive you may be able to use any grounded or non grounded pushbutton.
The critical dimensions are the plunger height above the PCB (from the included photos, about 7mm), and the PCB pitch (lead to lead distance for the 4 main ones, and the optional ground pin). Many of the electronic products you have around may have a similar push button for you to salvage. If the stem is longer than needed, trim it. If shorter, add some tape or glue a thin substance onto it. You can easily have this board up and running within an 10 minutes.
Note that I call the ground pin optional, but it does serve a function of reducing noise slightly, and ESD protection. However, I think it isnt strictly necessary in this application.
Bonus info: the buttons might not need to be replaced. Switches have metal inside them, which corrodes with time. This is why any switch with a datasheet will tell you about the minimum load (read more here, in the microload section). Any load bigger than specified (so, a larger current), would be sufficient to keep the switch operational. However, some designers ignore this. I am unsure about this, but I think if you pass a small current, for example turning on a 20 mA LED, you may be able to break down the oxide layer and allow the button to work a while longer. (If you know more about this, please comment and correct me. Thank you)