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Yes, you can allow a PC to take and use the frosty orb.

In order to use it, they will have to destroy their own eye for power. You should immediately consider the effect on their psyche of both contemplating and carrying out such an action.

The orb does not provide sight, so the one-eyed disadvantages will be appropriate. The fog will improve concealment in some circumstances and make the PC a more visible target in others. The orb doesn't offer cold resistance, so it is appropriate that cold environmental effects will be worse for the PC. On the other hand, they might be happier in a desert.

Any character who has experience with these cultists will assume that the PC is a cultist and therefore the whole party is.

As a general guideline, games are improved by side-effects of player decisions much more than they are ruined by power imbalance.

Finally, you should think about the Head of Vecna story: https://www.rpglibrary.org/articles/storytelling/headofvecna.php