Many books written by (early) Achronim are based on a particular shittah usually sourced in one or more of the Rishonim. This can be true both with regards to actual halachos or lists of mitzvos and of hashkofah books. Did the Ramchal have a particular shittah that he subscribed to which could be seen as a source for his book Derech Hashem? Also, can the Ramchal be said to have generally subscribed to a particular shittah in the Rishonim?
I was a bit confused by your question because 'shitah' is usually a term used in learning halakhah but Derekh Hashem isn't a book of halakhah, i.e., what to do, but rather a presentation of a conceptual framework for understanding religion and the world. As far as the sources of the ideas: they come in large part out the Ramhal's engagement with the writings of the Ari and his working out a way of understanding them. The Ramhal took the writings of the Ari to be metaphorical and sought to penetrate to inner meanings, concerning the workings of providence and the processes of spiritual life. He works out these metaphorical interpretations in his other books, like 'Klalim Rishonim', 'Kla"h Pithey Hokhmah', and 'Adir Ba-Marom'. Then, in Derekh Hashem, the Ramhal presents just the conclusions of his interpretative enterprise, without reference to the original language of the Ari and without explicitly using any kabbalistic terminology at all. So the content comes out of the Kabbalah of the Ari, but it's a personal understanding or re-interpretation of that body of teachings, and is stripped of its original terminology.
While the influence of Lurianic Kabbala is certainly there, a lot of the ideas in Derech Hashem can be traced back to R' Saadia Gaon's "Emunot v'Deot" "Emunot v'Deot" was the first attempt to present Jewish thought in a systematic framework, which is exactly what "Derech Hashem" does. Moreover, some of the actual content of Emunot v'Deot and Derech Hashem agree. I highly recommend the Hacket publishing version of "Emunot v'Deot" as a starting point to compare the two works, since the Hacket version presents selections from R' Saadia Gaon's philosophy.