Ordinarily, no. At a Catholic Mass, Holy Communion is for Catholics; the U.S. bishops' guidelines say members of churches not yet fully united with the Catholic Church "are ordinarily not admitted to Holy Communion." Canon 844 makes two exceptions: Orthodox Christians who ask on their own and are properly disposed, and other baptized Christians in danger of death or another grave necessity, under strict conditions. Everyone is welcome at the Mass itself.
The reason is what Communion means. The bishops' guidelines, printed in many missalettes, welcome fellow Christians to the celebration as brothers and sisters, and explain that Catholics hold the Eucharist to be a sign of a oneness of faith, life and worship that divided Christians do not yet share. Receiving together would say something that is not yet true.
The law itself is canon 844. Catholic ministers give the sacraments licitly to Catholics alone (§1). Members of the Eastern Churches not in full communion, and of churches the Apostolic See judges to be in the same condition, may receive the Eucharist if they ask on their own and are properly disposed (§3); the guidelines add that members of the Orthodox Churches, the Assyrian Church of the East and the Polish National Catholic Church are urged to respect the discipline of their own Churches. Other baptized Christians may receive in danger of death, or in another grave necessity in the judgment of the diocesan bishop or the conference of bishops, and only if they cannot approach a minister of their own community, ask on their own, manifest Catholic faith in the sacrament and are properly disposed (§4). Canon 842 adds that a person who is not baptized cannot receive any other sacrament.
Good to know on a pilgrimage: Mass is usually celebrated every day, so this comes up every day, and it is entirely ordinary to stay in your seat while others go forward. Nobody is checked or questioned. The grave-necessity exception is a judgment for the bishop, not something a tour leader or the priest at the altar settles on the spot, so if you think your situation is unusual, speak with the priest privately beforehand.
Related questions, answered
What should I do at Communion time if I am not receiving?
Stay in the pew, or, where it is the custom, come forward with arms crossed over your chest for a blessing. The blessing is a common practice and not a rule of the liturgy, so follow what the local congregation does. The bishops' guidelines ask everyone who is not receiving to express in their hearts a prayerful desire for unity with the Lord Jesus and with one another.
Source: USCCB: Guidelines for the Reception of Communion
Can an Orthodox Christian receive Communion at a Catholic Mass?
Catholic law permits it: canon 844 §3 allows a Catholic minister to give Communion to Orthodox Christians who ask on their own and are properly disposed. The U.S. bishops' guidelines urge Orthodox Christians to respect the discipline of their own Churches, which generally do not permit it, so an Orthodox pilgrim should follow the guidance of his or her own priest.
Source: Code of Canon Law, canons 842 and 844USCCB: Guidelines for the Reception of Communion
Can a non-Catholic attend Mass?
Yes. Mass is public worship and no one is checked at the door. The bishops' guidelines welcome other Christians and also "those who do not share our faith in Jesus Christ," asking all to pray for peace and the unity of the human family. A guest may sit, stand and kneel with the congregation or simply remain seated.
Source: USCCB: Guidelines for the Reception of Communion
