What is the role of a permanent lector at a Traditional Latin Mass?

First and foremost:

Furthermore, by virtue of its character of special law, within its own area, the Motu Proprio Summorum Pontificum derogates from those provisions of law, connected with the sacred Rites, promulgated from 1962 onwards and incompatible with the rubrics of the liturgical books in effect in 1962.

So the motu proprio of Francis doesn't change anything from what lectors already were permitted to do. Lay men have been at least theoretically possible as instituted lectors since 1975 under Paul VI. In practice, only seminarians routinely received those ministries (with few exceptions, namely the Diocese of Lincoln, Nebraska).

Whether an instituted lector (post-1975) is equivalent to an ordained lector (pre-1975) is somewhat an academic and controversial question, but assuming they are, then an instituted lector may chant the Epistle at Sung Mass, chant the Lesson when there are more than 2 (e.g. Ember Days) and chant the readings at the Solemn Good Friday Liturgy and the Divine Office (especially Matins).

What is the role of a permanent lector at a Traditional Latin Mass?

Both lectors and acolytes were known as minor orders and as such were steppingstones towards priestly ordination.

A lector (reader) in the West is a clerk having the second of the four minor orders. In all Eastern Churches also, readers are ordained to a minor order preparatory to the diaconate. The primary reason for a special class of readers was the need of some persons sufficiently educated to be able to read the books in church, for the Christians continued the Jewish practice of reading the Sacred Books publicly. The first mention of a Christian liturgical reader is by Justin Martyr (d. about 165) in I Apol., lxvii, 3, 4. The homily known as "II Clem. ad Corinthios" also contains a reference to a lector, anaginoskon (xix, 1). The position of reader was honourable and dignified. It involved a higher standard of education than that of most offices. Although Justin says that the bishop preached the sermon, it appears that the reader himself often went on to expound what he had read. As the idea obtained that a special blessing and dedication should be given to everyone who performs an office for the Church, the reader too was instituted by prayers and some ceremony. Readers were blessed and set apart, as were the fossores who dug graves, the notarii who kept registers, and widows. All the group of rituals that depend on the "Apostolic Constitutions" contain the rite of ordaining readers. "Apost. Const.", vii, xxii, tells the bishop to ordain a reader by laying on his hand and saying a prayer, which is given. The derived documents however forbid an imposition of hands. ("Epitome Const. Ap.", xiii; Funk, "Didascalia", Paderborn, 1905, II, p. 82; see also the "Egyptian Church Order", V, ib., p. 105).

During the first centuries all the lessons in the liturgy, including the Epistle and Gospel, were read by the lector. Cornelius I (251-53) in a letter to Fabius of Antioch mentions that the Church of Rome has forty-two acolytes and fifty-two exorcists, readers and doorkeepers. (Denzinger, "Enchiridion", n. 45). In the fourth century in Africa the Church of Cirta had four priests, three deacons, four subdeacons, and seven readers. The account of the persecution ("Gesta apud Zenophilum" printed in the appendix to Optatus of Mileve in the Vienna edition of "Corp. Script. eccl. lat.", XXVI, 185-97) describes how the readers kept the sacred books which the magistrate demanded to be given up (p. 187). An old set of Western canons, ascribed (wrongly) to a supposed Council of Carthage in 398, but really of the sixth century, gives forms for all ordinations. Canon 8 is about our subject: "When a reader is ordained let the bishop speak about him (faciat de illo verbum) to the people, pointing out his faith and life and skill. After this, while the people look on, let him give him the book from which he is to read, saying to him: Receive this and be the spokesman (relator) of the word of God and you shall have, if you do your work faithfully and usefully, a part with those who have administered the word of God" (Denzinger, op. cit., n. 156). But gradually the lectorate lost all importance. The deacon obtained the office of reading the Gospel; in the West the Epistle became the privilege of the subdeacon. In the Eastern Churches this and other lessons are still supposed to be read by a lector, but everywhere his office (as all minor orders) may be supplied by a layman. The lector is still mentioned twice in the Roman Missal. In the rubrics at the beginning it is said that if Mass be sung without deacon and subdeacon a lector wearing a surplice may sing the Epistle in the usual place; but at the end he does not kiss the celebrant's hand (Ritus celebr. Missam", vi, 8). On Good Friday the morning service begins with a prophecy read by a lector at the place where the Epistle is usually read (first rubric on Good Friday).

Everywhere the order of reader has become merely a stepping-stone to major orders, and a memory of early days. In the Roman Rite in is the second minor order (Ostiarius, Lector, Exorcista, Acolythus). The minor orders are conferred during Mass after the first Lesson; but they may be given apart from Mass, on Sundays or doubles, in the morning. The lectorate involves no obligation of celibacy or of any other kind. The Byzantine Office will be found in the "Euchologion" (Euchologion to mega, Venetian 8th edition, 1898, pp. 186-87). The Armenians (Gregorian and Uniate) have adopted the Roman system of four minor orders exactly. Their rite of ordaining a reader also consists essentially in handing to him the book of the Epistles. Lector

As for acolytes the Catholic Encyclopedia says the following:

An acolyte is a cleric promoted to the fourth and highest minor order in the Latin Church, ranking next to a subdeacon. The chief offices of an acolyte are to light the candles on the altar, to carry them in procession, and during the solemn singing of the Gospel; to prepare wine and water for the sacrifice of the Mass; and to assist the sacred ministers at the Mass, and other public services of the Church. In the ordination of an acolyte the bishop presents him with a candle, extinguished, and an empty cruet, using appropriate words expressive of these duties. Altar boys are often designated as acolytes and perform the duties of such. The duties of the acolyte in Catholic liturgical services are fully described in the manuals of liturgy, e.g. Pio Matinucci, "Manuale Sacrarum Caeremoniarum" (Rome, 1880), VI, 625; and De Herdt, "Sacrae Liturgiae Praxis" (Louvain, 1889), II, 28-39.

According to the ancient discipline of the Roman Church the order of acolyte was conferred as the candidate approached adolescence, about the age of twenty, as the decree of Pope Siricius (385) to Himerius, Bishop of Tarragona, in Spain, was interpreted (P.L., XIII, 1142). Five years were to elapse before an acolyte could receive subdeaconship. Pope Zosimus reduced (418) this term to four years. The Council of Trent leaves to the judgment of bishops to determine what space should elapse between the conferring of the acolythate and subdeaconship; it is also interesting to note, with Dr. Probst (Kirchenlex., I, 385), that the Council's desire (Sess. XXIII, c. 17, de ref.) concerning the performance of ministerial services exclusively by minor-order clerics was never fulfilled. In ancient ecclesiastical Rome there was no solemn ordination of acolytes. At communion-time in any ordinary Mass, even when it was not stational, the candidate approached the Pope, or in his absence, one of the bishops of the pontifical court. At an earlier moment of the Mass he had been vested with the stole and the chasuble. Holding in his arms a linen bag (porrigitur in ulnas ejus sacculus super planetam; a symbol of the highest function of these clerics, that of carrying, as stated above, the consecrated hosts) he prostrated himself while the Pontiff pronounced over him a simple blessing (Mabillon, op. Cit., II, 85, ed. Paris, 1724). It may be well to mention here the two prayers of the ancient Roman Mass-book known as the "Sacramentarium Gregorianum" (Mabillon, Lit. Rom. Vetus, II, 407), said by the Pontiff over the acolyte, and the first of which is identical with that of the actual Roman Pontifical "Domine, sancte Pater, aeterne Deus, qui ad Moysen et Aaron locutus es," etc.

According to the aforementioned "Statuta Ecclesiae Antiqua," which give us the ritual usage of the most important churches in Gaul about the year 500, the candidate for acolyte was first instructed by the bishop in the duties of his office, and then a candlestick, with a candle extinguished, was placed in his hand by the archdeacon, as a sign that the lights of the church would be in his care; moreover, an empty cruet was given him, symbolical of his office of presenting wine and water at the altar for the holy sacrifice. A short blessing followed. Acolyte