Objection 1. It would seem that those who are subject to another's power are not hindered from taking vows. The lesser bond is surpassed by the greater. Now the obligation of one man subject to another is a lesser bond than a vow whereby one is under an obligation to God. Therefore those who are subject to another's power are not hindered from taking vows.
Objection 2. Further, children are under their parents' power. Yet children may make religious profession even without the consent of their parents. Therefore one is not hindered from taking vows, through being subject to another's power.
Objection 3. Further, to do is more than to promise. But religious who are under the power of their superiors can do certain things such as to say some psalms, or abstain from certain things. Much more therefore seemingly can they promise such things to God by means of vows.
Objection 4. Further, whoever does what he cannot do lawfully sins. But subjects do not sin by taking vows, since nowhere do we find this forbidden. Therefore it would seem that they can lawfully take vows.
On the contrary, It is commanded (Numbers 30:4-6) that "if a woman vow any thing... being in her father's house, and yet but a girl in age," she is not bound by the vow, unless her father consent: and the same is said there (Numbers 30:7-9) of the woman that has a husband. Therefore in like manner other persons that are subject to another's power cannot bind themselves by vow.
I answer that, As stated above (Article 1), a vow is a promise made to God. Now no man can firmly bind himself by a promise to do what is in another's power, but only to that which is entirely in his own power. Now whoever is subject to another, as to the matter wherein he is subject to him, it does not lie in his power to do as he will, but it depends on the will of the other. And therefore without the consent of his superior he cannot bind himself firmly by a vow in those matters wherein he is subject to another.
Reply to Objection 1. Nothing but what is virtuous can be the subject of a promise made to God, as stated above (Article 2). Now it is contrary to virtue for a man to offer to God that which belongs to another, as stated above (Question 86, Article 3). Hence the conditions necessary for a vow are not altogether ensured, when a man who is under another's power vows that which is in that other's power, except under the condition that he whose power it concerns does not gainsay it.
Reply to Objection 2. As soon as a man comes of age, if he be a freeman he is in his own power in all matters concerning his person, for instance with regard to binding himself by vow to enter religion, or with regard to contracting marriage. But he is not in his own power as regards the arrangements of the household, so that in these matters he cannot vow anything that shall be valid without the consent of his father.
A slave, through being in his master's power, even as regards his personal deeds, cannot bind himself by vow to enter religion, since this would withdraw him from his master's service.
Reply to Objection 3. A religious is subject to his superior as to his actions connected with his profession of his rule. Wherefore even though one may be able to do something now and then, when one is not being occupied with other things by one's superior, yet since there is no time when his superior cannot occupy him with something, no vow of a religious stands without the consent of his superior, as neither does the vow of a girl while in (her father's) house without his consent; nor of a wife, without the consent of her husband.
Reply to Objection 4. Although the vow of one who is subject to another's power does not stand without the consent of the one to whom he is subject, he does not sin by vowing; because his vow is understood to contain the requisite condition, providing, namely, that his superior approve or do not gainsay it.
Source: Summa Theologica (New Advent)