The answer is yes. Unfortunately, I can't really say that much beyond that. Looking on JSTOR, the only article I could find of relevance was Mormonism and the Chinese State: Becoming an official church in the People's Republic of China? by Pierre Vendassi and Will Thornely in China Perspectives, No. 1 (97) (2014), and that only briefly referenced Taiping-era Mormon activity, citing two sources for brief Mormon missions in the early 1850s:
The latter is currently inaccessible to me (having neither a Google Books preview nor 'search this book' function) so I cannot help you with regards to what happened on these missions beyond what the LDS news article reads, which is that a combination of dangers in the interior and difficulty in actually learning Chinese meant that the only Mormon mission ended after just two months.
So yes, there are apparently records. However, I can't actually help you in finding out what they say; I can only inform you that they do exist somewhere. But, just for fun, I had a search of 'mormon taiping' into Google Books and saw what came up. Somewhat unsurprisingly, the top four results are about Taiping theology in relation to Mormonism, which is as good a segue as any for me to start talking about notable similarities in some of their theological precepts.
Among these is the idea that scripture as had existed for most Christians was in some way flawed. Hong Xiuquan, the de jure Taiping leader, spent much of his later life in Nanjing publishing the Bible, but editing various passages. The examples chosen by Jonathan Spence in God's Chinese Son are especially interesting – Hong cut out Lot's rape by his daughters, modified the Jacob and Esau story so that, instead of deceiving his father Isaac, Jacob was instead rewarded for acts of filial piety, and, instead of drinking wine, he and Isaac drank broth instead. Indeed, all references to Biblical figures (other than evil ones) drinking wine were expunged. Unlike the Mormons, however, scripture was never extended, only modified into a more perfect form.
The idea of a continuous revelation is somewhat of a feature of Taiping theology as well. Yang Xiuqing made liberal use of his status as God's voice both to enact laws and to compel people to do his bidding, (see Deng Siyu, Historiography of the Taiping Rebellion (1962) ch.1 for one such example) whilst Hong Xiuquan's dreams formed the basis of the Taiping mission, and would be continually updated. (See Jonathan Spence, The Taiping Vision of a Christian China (1996)).
Of course, other things are less linked between the two. As far as I'm aware, the Taiping had very little in the way of cosmological doctrine, and rejected Trinitarianism – God the Heavenly Father was the only truly divine being, and Jesus was merely his eldest son. There are more differences, of course, but not being massively aware of Mormonism's tenets beyond the basics, I cannot help you with much more than that.
Once again, sorry for not being too helpful.