No, they do not stack.
The Size Stacking FAQ states:
Size increases and effective size increases: How does damage work if I have various effects that change my actual size, my effective size, and my damage dice?
As per the rules on size changes, size changes do not stack, so if you have multiple size changing effects (for instance an effect that increases your size by one step and another that increases your size by two steps), only the largest applies. The same is true of effective size increases (which includes “deal damage as if they were one size category larger than they actually are,” “your damage die type increases by one step,” and similar language). They don’t stack with each other, just take the biggest one. However, you can have one of each and they do work together (for example, enlarge person increasing your actual size to Large and a bashing shield increasing your shield’s effective size by two steps, for a total of 2d6 damage).
Improvised Weapon Mastery states:
Increase the amount of damage dealt by the improvised weapon by one step
Which would fall under the similar language portion of the FAQ for effective size increases.
Given that Shikigami Style states:
While in this style, you deal damage with improvised weapons as if they were one size category larger. For every style feat you have that lists Shikigami Style as a prerequisite, treat the improvised weapon as an additional size category larger, to a maximum of three sizes larger than its actual size.
The clause in the FAQ about taking the highest effective size increase applies, and you would take the larger of the two, which will always be shikigami style since it will be the same or larger than improvised weapon mastery.