The Torah itself does not specify who reaps the grain, but it does state that “an omer of the first-fruits of your harvest” is brought “to the priest,” so we may deduce that it was not the priest who reaped the grain.
1 Lev. 23:10
The process itself is elaborated at length in the Talmud, which states,2
כיצד הן עושין שלוחי בית דין יוצאין מערב יום טוב ועושין אותן כריכות במחובר לקרקע כדי שיהא נוח לקצור כל העיירות הסמוכות לשם מתכנסות לשם כדי שיהא נקצר בעסק גדול כיון שהחשיכה אומר להן בא השמש אומר הין בא השמש אומר הין מגל זו אומר הין מגל זו אומר הין קופה זו אומר הין קופה זו אומר הין בשבת אומר להן שבת זו אמר הין שבת זו אמר הין אקצור והם אומרים לו קצור אקצור והם אומרים לו קצור שלש פעמים על כל דבר ודבר והן אומרים לו הין הין הין כל כך למה <לי> מפני הבייתוסים שהיו אומרים אין קצירת העומר במוצאי יו"ט
How do they do [it]? Delegates of the Beit Din go forth on the eve of the festival day (Yom Tov) and make bundles while [the sheaves] are connected to the ground so that it will be easy to reap. All the residents of the adjacent towns assembled there, so that it will be reaped with great affair. As soon as it was dark, [the reaper] says to [the people], “Has the sun set?” [The people] say, “Yes!” “Has the sun set?” [The people] say, “Yes!” “This sickle?” [The people] say, “Yes!” “This sickle?” [The people] say, “Yes!”
“This basket?” [The people] say, “Yes!” “This basket.” [The people] say, “Yes!” On the Sabbath, he says to [the people], “This Sabbath?” [The people] say, “Yes!” [He says to the people], “Shall I reap?” And [the people] say to him, “Reap!” [He says to the people], “Shall I reap?” And [the people] say to him, “Reap!” [He asks a question] three times for every thing, and [the people] say to him, “Yes! Yes! Yes!” Why so? Because of the Boethusians (Sadduccees) who would say that the omer is not reaped on the conclusion of the festival day (i.e., after Nisan 15).
2 Seder Kodashim, Tractate Menachot, Chapter 6, Folio 65a, Mishna
Moshe ben Maimon (Maimonides) wrote,3
How was [the offering] brought? On the day before the festival of Pesach, the agents of the court would go out [to the field] and tie [the barley] into bundles while it was still attached to the ground so that it would be easy to reap. [On the evening after Pesach,] all [of the inhabitants] of all the neighboring villages would gather so that it would be reaped with much flourish. They would have three men reap three se'ah of barley in three baskets with three sickles.
When it became dark, the reapers would ask those standing [in attendance]: "Has the sun set?" They would answer: “Yes.” “Has the sun set?” They would answer: “Yes.” “Has the sun set?” They would answer: “Yes.” “Is this a sickle?” They would answer: “Yes.” “Is this a sickle?” They would answer: “Yes.” “Is this a sickle?” They would answer: “Yes.” “Is this a basket?” They would answer: “Yes.” “Is this a basket?” They would answer: “Yes.” “Is this a basket?” They would answer: “Yes.” If it was the Sabbath, they would ask: “Is it the Sabbath?” They would answer: “Yes.” “Is it the Sabbath?” They would answer: “Yes.” “Is it the Sabbath?” They would answer: “Yes.” Afterwards, they would ask: “Should I reap?” They would answer: “Yes.” “Should I reap?” They would answer: “Yes.” “Should I reap?” They would answer: “Yes.”
Three [questions and answers] were given regarding each matter. Why was all this necessary? Because of those who erred who departed from the community of Israel in the Second Temple [era]. They maintained that the Torah’s expression [Leviticus 23:11]: “From the day following the Sabbath” [should be understood literally, as referring to] the Sabbath of the week. Nevertheless, according to the Oral Tradition, [our Sages] derived that the intent is not the Sabbath, but the festival. And so, was understood at all times by the prophets and the Sanhedrin in every generation. They would have the omer waved on the sixteenth of Nisan whether it fell during the week or on the Sabbath.
3 Mishneh Torah, Sefer Avoda, Temidin uMusafin, Ch. 7, Halakha 11
Alfred Edersheim wrote,4
When the time for cutting the sheaf had arrived, that is, on the evening of the 15th of Nisan (even though it were a Sabbath), just as the sun went down, three men, each with a sickle and basket, formally set to work.
But in order clearly to bring out all that was distinctive in the ceremony, they first asked of the bystanders three times each of these questions: ‘Has the sun gone down?’ ‘With this sickle?’ ‘Into this basket?’ ‘On this Sabbath (or first Passover-day)?’—and, lastly, ‘Shall I reap?’ Having each time been answered in the affirmative, they cut down barley to the amount of one ephah, or ten omers, or three seahs, which is equal to about three pecks and three pints of our English measure. The ears were brought into the Court of the Temple...
4 Edersheim, pp. 223–224
The Jewish Encyclopedia notes,5
The reaping was done with much ceremony. Messengers, sent by the bet din to the chosen field on the day preceding the Passover Feast, drew the heads of the stalks together in sheaves and tied them in order to facilitate the work of the reapers. Then when the hour for gathering came the reapers thrice asked permission to reap; this was done in order to impress upon the Boethusians that this was the proper time for the gathering of the ‘omer (Men. vi. 3).
5 Vol. 9, ‘Omer, p. 399, left column
Long story short, the Beit Din assigns a trustworthy delegate (or three delegates) with the task of reaping the grain on Nisan 16 and bringing it to the Temple.
The Feast of Unleavened Bread is seven days long, so unless I am mistaken, one could appear at the Temple at any point during that period to fulfill the mitzvah. Furthermore, one did not have to remain in Jerusalem during the entire seven days after fulfilling their obligation.
Edersheim, Alfred. The Temple—Its Ministry and Services, as They Were at the Time of Jesus Christ. London: The Religious Tract Society, 1874.
Moshe ben Maimon. Mishneh Torah. Trans. Touger, Eliyahu. Vol. 22. Brooklyn: Moznaim.
The Jewish Encyclopedia: A Descriptive Record of the History, Religion, Literature, and Customs of the Jewish People from the Earliest Times to the Present Day. Ed. Greenstone, Julius H. Vol. 9. New York: Funk & Wagnalls, 1907.