Which period in Chinese history produced the most chengyu?
I'll answer based on the article provided by @Stan. From what I can tell, it is a transcription from a 1999 article that appeared in 《语文建设》, the raw data of which came from 汉语成语考释词典 by 刘洁修, 1989.

Period                           | Number | Percent
---------------------------------+--------+--------
春秋以前 Before Spring and Autumn |     88 |    1.21
春秋 Spring and Autumn           |    320 |    4.41
战国 Warring States              |    232 |    3.19
秦 Qin                           |      1 |    0.01
两汉 Han                         |    479 |    6.59
三国 Three Kingdoms              |     96 |    1.32
两晋 Jin                         |    198 |    2.73
南朝 Southern Dynasties          |    270 |    3.72
北朝 Northern Dynasties          |     71 |    0.98
隋 Sui                           |     16 |    0.22
唐 Tang                          |    711 |    9.79
五代 Five Dynasties              |     75 |    1.03
两宋 Song                        |   1251 |   17.22
金 Jin (Jurchen)                 |     36 |    0.50
元 Yuan                          |    401 |    5.52
明 Ming                          |    854 |   11.76
清 Qing                          |   1494 |   20.57
Modern/Unknown                   |    671 |    9.24




But these numbers aren't all directly comparable; some dynasties such as Qin only existed very briefly, others such as North/South dynasties co-existed. I've therefore combined some and produced a graph showing number of chengyu per year:



Caveats


For the number of years "before Qin", I started counting from the Zhou dynasty.
It's probably unfair to show the "Modern/Unknown" category as-is; strictly speaking, the source material only categorises up to Qing dynasty, but according to the source, "most" of those in this category are probably modern.


On determining the source of a chengyu

It's not easy determining the source of a chengyu, since like the rest of the language, it evolves and even changes meaning over time. The method used in these statistics is the first record of a chengyu in its modern form. This would necessarily bias the data towards more recent ages, even though those chengyu either had different forms earlier, or were modern chengyu based on older origins.

To use an example from the article, the chengyu 镜花水月 is recorded as appearing in Ming, however it had appearances earlier:


In Jin, there is the passage "如镜中像、水中月，见如有色，而无触等，则非色也"
In Tang/Song, there is "水月镜像"; at this stage it is not quite chengyu as its form has not yet stabilised


However, even in Ming, this chengyu had a different meaning to its modern meaning. Its modern meaning only appeared in Qing.

Therefore, the results may differ quite a lot depending on how you determine the origin of a chengyu.