I am going to extrapolate from the behavior of the California fathers under the assumption that they followed canon law. The answer depends on which sort of officiant is involved. It turns out there are lots of examples of an Officiant who is not a priest — they could even be women — for provisional (partial) baptisms. Then the sacrament could be completed later by a priest, the Supli Ceremonia Officiant. I quote from the guide to the database:
There are two types of baptism: standard or provisional. If the baptism was a standard baptism — one in which the ego was healthy and able to participate — it was coded as normal ... the padres also baptized individuals who they feared were in danger of imminent death or were otherwise unable to travel to the mission due to ill health.
Usually the officiant is a missionary, but in the cases of provisional baptisms, the officiant could have been a layperson.
In many provisional baptisms no godparents were assigned at the time of baptism. At a later date, if the ego survived, the Franciscans would execute the complete rite of baptism, stating in the record that they had “supli las ceremonias,” or in a crude translation, supplied the full ceremonies. It should be noted that this was not a second baptism, but the completion of the original baptism. It was at this time that the godparents might be assigned.
The database also contains Nuptial Officiants and Dispensation Officiants for other specific cases with regard to marriage.