Do Catholics have to request a religious exception for matters of conscience from their parish priest?

Basically, you are asking two separate questions here:

Do Catholics have to request a religious exception for matters of conscience from their parish priest?

Is there any reason that an individual Catholic shouldn't seek a religious exemption from a member of the clergy who shares their concerns with the nature of the vaccines, even if is not their parish priest?

Alas, I am relatively certain a dispensation is only for religious matters. Not sure a vaccine exemption actually qualifies.

To request a religious exemption for matters of conscience is basically asking for a dispensation, which is generally given by the Local Ordinary of Bishop of a particular diocese or his duly instituted delegate.

Can. 87 §1. A diocesan bishop, whenever he judges that it contributes to their spiritual good, is able to dispense the faithful from universal and particular disciplinary laws issued for his territory or his subjects by the supreme authority of the Church. He is not able to dispense, however, from procedural or penal laws nor from those whose dispensation is specially reserved to the Apostolic See or some other authority.

There are several forms of dispensations or exemptions as you put it.

Kinds of Dispensation.—(a) A dispensation may be explicit, tacit, or implicit, according as it is manifested by a positive act, or by silence under circumstances amounting to acquiescence, or solely by its connection with another positive act that presupposes the dispensation. It may be granted in foro interno, or in foro extern, according as it affects only the personal conscience, or conscience and the community at large. Although dispensations in foro interno are used for secret cases, they are also granted in public cases; hence they must not be identified with dispensations in case occulto. A dispensation may be either direct or indirect, according as it affects the law directly, by suspending its operation, or indirectly, by modifying the object of the law in such a way as to withdraw it from the latter’s control. For instance, when a dispensation is granted from the matrimonial impediment of a vow, the pope remits the obligation resulting from the promise made to God, consequently also the impediment it raised against marriage. A dispensation may be in forma gratiosa, in forma commissa, or in forma commissa mixta. Those of the first class need no execution, but contain a dispensation granted ipso facto by the superior in the act of sending it. Those of the second class give jurisdiction to the person named as executor of the dispensation, if he should consider it advisable; they are, therefore, favors to be granted. Those of the third class command the executor to deliver the dispensation if he can verify the accuracy of the facts for which such dispensation is asked; they seem, therefore, to contain a favor already granted. From the respective nature of each of these forms of dispensation result certain important consequences that affect delegation, obreption, and revocation in the matter of dispensations.

The power of dispensation rests in the following persons:

(A) **The Pope. — He cannot of his own right dispense from the Divine law (either natural or positive). When he does dispense, e.g. from vows, oaths, unconsummated marriages, he does so by derived power communicated to him as Vicar of Christ, and the limits of which he determines by his magisterium, or authoritative teaching power. There is some diversity of opinion as to the nature of the pope’s dispensing power in this respect; it is generally held that it operates by way of indirect dispensation: that is, by virtue of his power over the wills of the faithful the pope, acting in the name of God, remits for them an obligation resulting from their deliberate consent, and therewith the consequences that by natural or positive Divine law flowed from such obligation. The pope, of his own right, has full power to dispense from all ecclesiastical laws, whether universal or particular, even from the disciplinary decrees of ecumenical councils. Such authority is consequent on his primacy and the fullness of his immediate jurisdiction. A part of this power, however, he usually communicates to the Roman Congregations.

(B) The Bishop. — Of his ordinary right, the bishop can dispense from his own statutes and from those of his predecessors, even when promulgated in a diocesan synod (where he alone is legislator). From the other laws of the Church he cannot dispense of his own right. This is evident from the nature of dispensation and of diocesan jurisdiction. A principle maintained by some authors, viz. that the bishop can grant all dispensations which the pope has not reserved to himself, cannot be admitted. But by derived right either ordinary or delegated according to the terms of the grant) the bishop can dispense from those laws that expressly permit him to do so or from those for which he has received an indult to that effect. Moreover, by ordinary right, based on custom or the tacit consent of the Holy See, he may dispense: in a case where recourse to the Holy See is difficult and where delay would entail serious danger; in doubtful cases, especially when the doubt affects the necessity of the dispensation or the sufficiency of the motives; in cases of frequent occurrence but requiring dispensation, also in frequently occurring matters of minor importance; in decrees of national and provincial councils, although he may not pronounce a general decree to the contrary; in pontifical laws specially passed for his diocese. It should be always remembered that to fix the exact limit of these various powers legitimate custom and the interpretation of reputable authors must serve as guides. Superiors of exempt religious orders can grant to their subjects, individually, those dispensations from ecclesiastical laws which the bishop grants by his ordinary power. When there is question of the rules of their order they are bound to follow what is laid down in their constitutions.

(C) The Vicar-General. — He enjoys by virtue of his appointment the ordinary dispensing power of the bishop, also the delegated powers of the latter, i.e. those granted him not personally but as ordinary (according to present discipline, the pontifical faculties known as ordinary); exception is made, however, for those powers which require a special mandate like those of the chapter Liceat, for dealing with irregularities and secret cases. The vicar capitular likewise has all the dispensing power which the bishop has of his own right, or which has been delegated to him as ordinary.

(D) Parish Priest. — By his own ordinary right, founded on custom, he may dispense (but only in particular cases, and for individuals separately, not for a community or congregation) from the observance of fasting, abstinence, and Holy Days. He can also dispense, within his own territory, from the observance of diocesan statutes when the latter permit him to do so; the terms of these statutes usually declare the extent of such power, also whether it be ordinary or delegated. Dispensation being an act of jurisdiction, a superior can exercise it only over his own subjects, though as a general rule he can do so in their favor even outside his own territory. The bishop and the parish priest, except in circumstances governed by special enactments, acquire jurisdiction over a member of the faithful by reason of the domicile or quasi-domicile he or she has in a diocese or parish. Moreover, in their own territory they can use their dispensing power in respect of persons without fixed residence (vagi), probably also in respect of travellers temporarily resident in such territory. As a general rule he who has power to dispense others from certain obligations can also dispense himself. - Dispensation

As for the second part of this question, some bishops have already stated that parish priests are not to grant Religious Exemption from Covid-19 Vaccine Mandates:

Bishop Daniel Garcia of Monterey on Friday directed clergy not to provide letters of religious exemption for Catholics wishing not to receive a coronavirus vaccine.

“Pope Francis has highly encouraged the vaccination as an act of safeguarding one’s own health as well as that of others – pointing us to the importance of the common good, such a key element of Catholic Social Teaching,” Bishop Garcia wrote in his Aug. 13 letter, adding that “the Bishops of California have also encouraged vaccination for the same reasons.”

“For these reasons, I will not issue, and I have directed our clergy not to issue, any Letters of Religious exemption because it would contradict the clear objective teaching of the Catholic Church and the Holy See on this matter,” Bishop Garcia wrote.

States and employers across the country have rolled out Covid-19 vaccine mandates in recent weeks. Last week, California became the first state to require all teachers, both public and private, to get the vaccine or else submit to weekly testing. Healthcare workers in California will soon be required to take the vaccine as well, though that order does include a religious exemption.

Before and as Covid-19 vaccines began to roll out, some Catholics raised concerns about the drugs’ remote connection to aborted fetal tissue. Those produced by Pfizer and Moderna were tested on cell lines derived from elective abortions decades ago, while the vaccine created by Johnson & Johnson was directly produced using the cell lines.

The U.S. Conference of Catholic Bishops has since stated that all three vaccines approved for use in the United States are “morally acceptable” for use because of their remote connection with abortion, but if one has the ability to choose a vaccine, Pfizer or Moderna’s vaccines should be chosen over Johnson & Johnson’s.

In fact, Pope Francis urges people to receive Covid-19 vaccine, as seen in this YouTube video: Pope urges people to receive Covid-19 vaccine The Pope, and other prelates urge Covid vaccination as act of love! Pope Francis was joined by six other prelates—Archbishop José Gomez of Los Angeles, Cardinal Carlos Aguiar Retes of Mexico City, Cardinal Óscar Rodríguez Maradiaga of Honduras, Cardinal Cláudio Hummes of Brazil, Cardinal Gregorio Rosa Chávez of El Salvador, and Archbishop Héctor Miguel Cabrejos Vidarte of Peru—in encouraging vaccination.

Therefore there is no reason that an individual Catholic should seek a religious exemption from a member of the clergy who shares their concerns with the nature of the vaccines since this subject matter is as yet, not listed in those things as what a parish priest may dispense of at this moment in time for member of the faithful.

Canonically, it would be ordinary for the clergy who have jurisdiction over you to witness on your behalf. This would generally be your parish priest (pastor) or ordinary (diocesan bishop) or their legitimate delegates.

A secondary possibility would be a spiritual director who is often neither your pastor nor bishop.

For this particular case, there may be neither particular civil law nor canon law which requires only those priests to authorize exemption (in reality, this is more that the clergy asserts that you do hold those religious views necessary for exemption from the civil law).

A similar case could be drawn to a medical exemption. It would be typical to get one from a primary care physician or from a specialist under whose care you were. It would not be usual to ask your golfing buddy who happens to be a doctor to grant one -- and he may in fact run into ethical issues there anyways.