In matters such as this, one needs to be obedient to the bishop of his diocese, as his bishop is his superior. Now, what exactly that means will be nuanced. The bishop may only make a recommendation or request, and such does not necessarily need to be obeyed. If the bishop gives a just command, that needs to be obeyed, and a just command here is defined as a command which the giver has the authority to give. The bishop is not in this particular case commanding the faithful to commit sin, and he does have the authority to dictate the particulars regarding liturgical actions in his jurisdiction. Hence, if the cardinal actually gave a command (and did not merely express his personal opinion or desire, or merely make a request), then the faithful are bound to obey it (I'm editing this to correct a point of fact; the Holy See has in fact ruled that the faithful cannot be denied communion for kneeling, which would seem to indicate that the Cardinal does not in fact have the right to require standing, but I think that if we suppose he hypothetically does, this answer is still valuable for possible future conflicts).
As to the matter of your vow, we should examine canon 1194 of the 1983 code, which says "A vow ceases by the lapse of the time designated to fulfill the obligation, by a substantial change of the matter promised, by the absence of a condition on which the vow depends, by the absence of the purpose of the vow, by dispensation, or by commutation." In my judgement, if you live in Chicago and Cardinal Cupich has really given a command (and we should take care to verify that he has), then there is now an absence of a condition on which your vow depends, namely, that you have the religious right to choose to kneel for communion. If your bishop has taken that right away, you no longer have it, and having it is a necessary condition for the fulfillment of your vow. Hence, the vow has now ceased (assuming everything said above is really true).
Furthermore, as this is a private vow, you might be able to commute it yourself under Cano 1197, which says "The person who makes a private vow can commute the work promised by the vow into a better or equal good; however, one who has the power of dispensing according to the norm of can. 1196 can commute it into a lesser good." You could commute your vow to a higher good, and it may serve you well to seek counsel on this from a trusted pastor if you are considering this route. But, I think this is a secondary concern. In my estimation, the vow has ceased because it has become impossible to fulfill due to unforeseeable changes in circumstances (assuming the Cardinal really has given a command).
Finally, to the question of conscience, one must follow a well-formed conscience above all else, but a well-formed conscience will compel one to be obedient to the bishop. Canon law seems to make provision regarding your vow such that a crisis of conscience should not really exist.
On obeying sinful superiors
A greater explanation on canon law as it relates to private or personal vows