In current Catholic practice the rites for a dying person run in an order that was deliberately restored in the twentieth century: penance, then the anointing of the sick, then viaticum. Communion comes last, and the Church describes it as the last sacrament of the Christian life. That ordering had slipped in the medieval West, where the last rites were popularly understood to culminate in extreme unction, the anointing which by then had become almost exclusively a sacrament of the dying rather than of the sick. Putting communion back at the end changes what the sequence says: the final act is not a preparation applied to a person but a meal shared with them.
Who may bring it follows from the same emphasis. A priest is the ordinary minister, but a deacon or another authorised minister of communion may carry the sacrament to a dying person where no priest can come, so the availability of viaticum does not depend on the availability of a priest in the way that anointing does.
Other traditions do the substance without the vocabulary. Orthodox parishes commune the dying from the sacrament reserved for the purpose. Anglican and Lutheran churches bring communion to the sick and dying as a normal pastoral office, and Anglican usage does sometimes keep the word. In the Reformed and free churches, where nothing is reserved and the elements are ordinary bread and wine again after the service, a minister and one or two members of the congregation bring the whole action to the bedside, which is understood as the congregation's table extended to a member who cannot come to it rather than as a distinct rite.