What order Tuck might have belonged to depends upon the span of years involved.
Benedict of Nursia (ca. 480-548) founded what we think of as monasticism in the Catholic tradition. Francis Bernadone founded what became the Franciscans in the winter of 1206/07; the Order of Friars Minor were giving papal approval by Innocent III and the Rule of the order was confirmed by Honorius III in 1223.
Since "King John" is probably John Lackland (ruled 1199 to 1216) it is faintly possible that Tuck was a Franciscan. Far more likely that he was a Benedictine, which order came to England with Augustine of Canterbury in 597. However, the monasteries did their best (and it was quite good!) to control the lives of the inmates -- it is unlikely that Tuck was a Benedictine simply because of the control exercised by the Abbot. The Cistercians were similar to the Benedictines in this regard (and still are).
This leaves the the "mendicant friars," of which four orders -- Franciscans, Dominicans, Augustinians, and Carmelites -- were recognized by the Second Council of Lyons in 1274.
By 1398, the approximate date of the Canterbury Tales, friars -- especially the mendicant friars -- had a reputation for various forms of sin and worldliness.
Given all this, yes, Friar Tuck was probably a layman and not an ordained priest. I suspect that the term "Friar" may also have been a nickname (like "Little" John) attached to the man for some reason, perhaps because of his physical girth.