Here one could reinterpret elders and priests as plural possessive forms: “The manner of their elders’ and priests’ administering the flesh and blood of Christ”. (It is also possible that one could treat “elders and priests” as combinatory rather than segregatory, thus adding the apostrophe to only the last conjunct: “The manner of their elders and priests’ administering the flesh and blood of Christ”.) This possessive interpretation is indirectly supported by the language at the beginning of the following chapter: “The manner of administering the wine” (Moroni 5:1). In other words, in Moroni 4:1 we have the equivalent “The manner of … administering the flesh and blood of Christ unto the church”.
The change to the possessive would make the gerundive phrase more nominal than verbal. We can perhaps see the potential difference more clearly if we replaced the phrase “their elders and priests” with a third person plural pronoun, either them or their: the more nominal form would be “The manner of their administering the flesh and blood of Christ” while the more verbal form would be “The manner of them administering the flesh and blood of Christ”. For phrases of this type, the Book of Mormon text prefers the nominal forms, as explained under Mosiah 16:3. Adding apostrophes here in Moroni 4:1 (or perhaps only one apostrophe) would draw unnecessary attention to a distinction that makes little difference since the text here has plural nouns rather than pronouns. The critical text will therefore leave the text as it is, without adding any apostrophe(s) to the phrase “their elders and priests”.
Summary: Maintain in Moroni 4:1 the object form of the noun phrase “their elders and priests”; changing this noun phrase to a possessive form by adding one or two apostrophes would create a rather odd-looking gerundive phrase.