First of all, it’s probably safe to assume that all lines are actually straight and not slightly curved. This allows you to reconstruct the springers , as the missing triangle is uniquely defined (see below).
Assuming that the numerical information you have is correct, it does not suffice to reconstruct the remaining voussoirs, because the base lengths and heights do not uniquely identify a trapezium:
What you can probably do is to assume that all voussoirs (except the springers) are isosceles trapezia, which would yield uniqueness. (You can probably also assume that contrary to the drawing you received, they are all identical.) In that case, you can reconstruct the shape step by step by placing voussoirs on each other, starting with the springers.
If everything works out and matches your given width and height, this would provide good evidence that your assumptions were correct.
As for the program you use for this, it really depends on what you have available and are used to. Assuming that you are not familiar with geometric construction programs or CAD programs, it is probably not worth the effort of learning them. Constructing the trapezia should be straightforward, and you only have to geometrically construct the springer, which goes like this (where x is the leg of the respective trapezium):