I find the General Instruction of the Roman Missal confusing on this point.
It has:
“119. … When there is an Entrance Procession, the following are also to be prepared: the Book of the Gospels; on Sundays and festive days, the thurible and the boat with incense, if incense is used; the cross to be carried in procession; and candlesticks with lighted candles.
A. MASS WITHOUT A DEACON
The Introductory Rites
120. Once the people have gathered, the priest and ministers, clad in the sacred vestments, go in procession to the altar in this order:
a. The thurifer carrying a thurible with burning incense, if incense is used;
b. The ministers who carry lighted candles, and between them an acolyte or other
minister with the cross;
c. The acolytes and the other ministers;
d. A lector, who may carry the Book of the Gospels (though not the Lectionary), which
should be slightly elevated;
e. The priest who is to celebrate the Mass.”
Here n. 119 seems to say explicitly that the procession is optional. But n. 120 seems to say that there will be a procession.
One interpretation is that there are different types of Mass. This includes “Mass At Which Only One Minister Participates”. For this type of Mass it has: “256. The priest approaches the altar and, after making a profound bow along with the minister, venerates the altar with a kiss and goes to the chair.” So for this type of Mass perhaps this is not called a procession.