Is it permissible to perform Istikhara through dreams, by saying, for example: "O Lord, if she is lying, let me dream of my sister, and if she is truthful, let me dream of her"?

If your friend is trustworthy, the default is to believe her. However, if she is known to lie, then verification is obligatory, based on the Almighty's saying: "O you who have believed, if there comes to you a disobedient one with information, investigate, lest you harm a people out of ignorance and become, over what you have done, regretful." What you are doing is not a correct way of verification, because dreams can be from Allah, from Satan, or from inner thoughts. Satan is keen on causing discord among people, so he might appear to you in the image of your friend, causing you to doubt her. Therefore, it is advisable to verify through another means.