What a “dispensation” is
First of all, a “dispensation” is a legal term, in which the competent authority relieves one of its subjects from having to follow a law (or part of a law) in a particular case. The Code of Canon Law describes dispensations as follows:
A dispensation, or the relaxation of a merely ecclesiastical law in a particular case, can be granted by those who possess executive power within the limits of their competence, as well as by those who have the power to dispense explicitly or implicitly either by the law itself or by legitimate delegation (Can. 85).
It is to be noted that a dispensation can be granted only for a strictly ecclesiastical law.
There is no general requirement that a dispensation be written, however Can. 37 says,
An administrative act which regards the external forum must be put in writing.
Hence, a dispensation that must be made public must be in writing. (A common dispensation of this kind, for example, is the one required for a Catholic to marry an unbaptized person.)
(The implication is that a merely disciplinary dispensation—say, from the obligation to fast on Ash Wednesday—could be given by word of mouth.)
Regarding the particular case in question
Regarding the matter in the original question (getting a vasectomy), the short answer is “no.” Vasectomies are a form of direct sterilization and mutilation, which cannot be justified under any circumstances:
Equally to be condemned, as the magisterium of the Church has affirmed on many occasions, is direct sterilization, whether of the man or of the woman, whether permanent or temporary (Humanae vitae, 14).
A bishop has no power whatsoever to set aside the natural law. (If a bishop ever gave that kind of advice, it would be advisable to notify the competent authority, either the nuncio of the country where he resides, or the Congregation for the Doctrine of the Faith directly.)
Moreover, the marriage “vows” (technically, the matrimonial consent) include the will to enter a stable, exclusive, monogamous, and fruitful union between a man and a woman. These are the essential properties of marriage, and they can never be set aside. As the Catechism puts it,
Unity, indissolubility, and openness to fertility are essential to marriage. Polygamy is incompatible with the unity of marriage; divorce separates what God has joined together; the refusal of fertility turns married life away from its “supreme gift,” the child (no. 1664, quoting Gaudium et Spes 50 § 1).
It may be justifiable to delay childbirth, even indefinitely (and a serious health problem on the woman’s part would justify it), but for that purpose a method that does not deliberately disrupt the fecundity of marriage must be used (i.e., natural family planning):
If therefore there are well-grounded reasons for spacing births, arising from the physical or psychological condition of husband or wife, or from external circumstances, the Church teaches that married people may then take advantage of the natural cycles immanent in the reproductive system and engage in marital intercourse only during those times that are infertile, thus controlling birth in a way which does not in the least offend the moral principles which We have just explained (Humanae vitae, 17)
It should be noted that since the time of Humanae vitae, there have been great advances in gynecology, which make it possible to use NFP with extremely high efficacy—for example with the Billings Ovulation Method or the Creighton Model. (When the health of the woman is in danger, then extra care should, naturally, be exercised to avoid pregnancy—and when that is done, the chance of pregnancy is very small.)
Regarding the attempted dispensation in question
More apropos to the original question, since giving dispensations from natural law is impossible (and attempting to so would most likely get him in trouble with the Curia), it is unlikely that a bishop who attempted to do so would leave a written record of it.
Moreover, the situation of a married couple (apart from the matrimony itself, which is necessarily in the external form) would in the internal forum (i.e., a private matter), and so it seems that any dispensations (if they were possible) would not need to be written to be official.
If the bishop (or other authority) were legitimately dispensing from a strictly ecclesiastical law, in a strictly private matter (e.g., regarding fasting in Lent), then no written record would be necessary.
If, on the other hand, the matter were public (in the external form), some kind of formal written documentation would be required (as in the case of disparity of cult in marriage).
However, in the case described (the vasectomy), a dispensation is impossible, and so the question is actually moot: the act was clearly both illicit and invalid.