On a certain level it's hard to answer this question with a lot of certainty, just based on what sort of documentary evidence we have. There's a lot of oracle bones from Shang times, but anything before that is a lot less certain. There are people who are skeptical that the Xia dynasty that's meant to have preceded the Shang even existed. The oracle bones offer a lot of insight into Shang thinking, although I want to be careful about the conclusions we draw from them, since we're piecing together lots of little fragments of thought and it's naturally hard to create a coherent picture of an overall philosophy that way. The other challenge is that writing about events in this period that we do have often date to later and come from times when notions like the Mandate were clearly accepted. First let's look at those writing and then to the oracle bones (I'll admit up front I'm not an expert on the oracle bones).

Zhou dynasty classics describing the Shang-Zhou transition already describe dynastic transition in terms of the Mandate. Here's a quote from the "Shao Announcement", a supposed record of the founding of a new Zhou capital during the transition period where the Duke of Zhou addresses a Shang audience (translation from the Columbia Sources of Chinese Tradition) "Heaven has rejected and ended the Mandate of this great state of Yin. Thus, although Yin has many former kings in Heaven, when their successor kings and successor people undertook their Mandate, in the end wise and good men lived in misery." You'll recognize this sort of language as being very much in the vein of the Mandate of Heaven justification. The Duke of Zhou is speaking about the Mandate as if his audience is familiar with it, does that suggest that the Mandate was already a key concept during Shang times? Perhaps, but not necessarily.

Another key aspect of the Shang-Zhou transition is Heaven surpassing Shangdi "the lord on high". The Shang had a deity, Shangdi, which fell out favor in Zhou times. In the early Zhou, Heaven was still a personified deity of sorts; Heaven would occasionally decide to do something actively and set out to do it. In later times Heaven came to be more abstract, representing the natural order, rather than a specific deity. There seems to be some competition between the two deity-like conceptions, sometimes Di will choose not to do something but Heaven will, although it's far from clear exactly how all the terms related back then. What does seem to have happened clearly, though, is the transition towards a broader, more abstract and universal conception of Heaven and the natural order.

With that in mind, the oracle bones definitely seem to suggest a theological/divine aspect to rule. The king had an active role in divination, and by the end was the key diviner. The king is therefore in a privileged position from his position to communicate with the dead. This connection is somewhat less clear than the Mandate would be in later thinking; the king would make predictions, and get it wrong sometimes after all. Still, there clearly was a theological/spiritual aspect to rule.