Seeing how Kaguya possesses both the rinne sharingan, which is the progenitor of the rinnegan, and the byakugan it is within the realm of possibilities that a byakugan user may awaken the rinnegan, but as a forehead third eye since they already have byakugan in both eyes and the original source of rinne and tensei chakra is Kaguya herself. But I would imagine that only select Hyuga or Moon Shinobi Otsutsuki clan members can do this depending on the purity of their blood that match either of Hamura's aspects (Hyuga or Moon Shinobi Otsutsuki) and/or whether or not Hamura's children have unique chakra like Indra and Asura that makeup the chakra of their parent and only them and their transmigrants can awaken the rinnegan/ rinne sharingan. This is an hypothetical conjecture since what you're asking has remained unanswered at the end of the Naruto series.
In an hypothetical scenario, it may be possible for an eligible Hyuga clan member to acquire Asura & Indra's chakra or chakra from Hagoromo himself and they may acquire a single forhead rinnegan. For the sake of logistics, lets say they acquire one purple rinnegan in their forehead. Why? Because there may be a component in Hamura's chakra that may mutate the rinnegan into the rinne sharingan and thus since we are talking about one third of said chakra they may acquire an incomplete rinne sharingan. For simplicity's sake let's say that an eligible Hyuga clan member can acquire a purple rinnegan and that an eligible Moon Shinobi Otsutsuki clan member can acquire a red rinnegan. When that missing 1/4th is acquired this may trigger the third eye to transform into its complete form, the rinne sharingan.
This hypothesis may connect with another hypothesis as to why Kaguya doesn't appear to have the tenseigan while Hamura has it. Perhaps by acquiring Hamura's chakra and the tenseigan and then acquiring the rinne aspect that the user previously lacked, said individual may awaken the third eye as the complete rinne sharingan, but as a consequence of its activation the tenseigan deactivates, because the tensei chakra now has a rinnegan to be processed by and thus in the absence of rinne chakra and the rinnegan the tensei chakra manifests as the tenseigan in the user's byakugan. This maybe the reason why Kaguya appears to lack the tenseigan while his son doesn't. I would also like to add that Kaguya's byakugan is considered a kekkei mora rather than a kekkei genkai and it may very well be for this reason. This is taking into account that Hagormo and Hamura each hold an aspect of Kaguya's chakra in their own and if both chakra were to merge, her chakra may likely mainfest in the individual. As already mentioned, Asura + Indra = Hagoromo as shown by Madara and unspecified Hyuga + unspecified Moon Shinobi Otsutsuki = Hamura as demonstrated by Toneri. So why wouldn't Hagoromo + Hamura = Kaguya? Let's also take into account that Hagoromo's rinnegan are purple and tomoeless while his mother's rinnegan is red and with (9) tomoe like the sharingan. The absence of the tensei component in Hagoromo may be the reason why there is a difference between his eyes and his mother's eye.
A possible counter arguement for this last paragraph would be that Kaguya can indeed activate the tenseigan and there is no relationship between the tenseigan and her rinne sharingan as explained above. She may have simply chosen not activate them during her fight with Naruto and Sasuke. It can be further argued that her tenseigan may be more advanced compared to her son's seeing as how her son's tenseigan seems to be able to evolve further by sealing other clan member's byakugan inside it. Being the source of all chakra her tenseigan could very well be more advance than her son's. However, this hypothesis is also debatable as there is no evidence that supports that she has the tenseigan.
Hope that this explaination makes sense and even if there is something that you may disagree with or that you don't like, because it's speculative in nature that at least this attempt of explaining an untestible unknown (until Kishi gets off his ass and tell us the answer) may get your imagination going so you may actually formulate your own explainations on this matter.