Here the printer’s manuscript reads became, while the 1830 edition reads become. The question, then, is whether the meaning is ‘the people began to be proud and to become vain’ (the reading of the 1830 edition and most editions) or ‘the people began to be proud and they became vain’ (the reading of the printer’s manuscript and RLDS editions since 1892).
Oliver Cowdery frequently wrote the letters a and o similarly, so it is quite possible that scribe 2 in the printer’s manuscript misinterpreted become in the original manuscript as became. But of course the opposite could have happened: the 1830 typesetter could have misread a became in the original manuscript as become. For other instances where become and became have been mixed up in the history of the text, see under Alma 13:9.
The best evidence for resolving this problem comes from usage elsewhere in the text. First of all, we have four examples in the text where the verb in the conjoined clause is in the past tense and thus agrees with a began in the preceding clause:
In these instances, the first clause refers to a continuous action, but the following conjoined clause refers to a completed specific action.
In contrast, there are 39 cases where the conjoined clause is in the infinitive form (there are examples with and without the repeated infinitival marker to), as in Helaman 16:12, where the to is not repeated: “the people began to be more hardened in iniquity and do more and more of that which was contrary to the commandments of God”. (For further discussion of this type, see under Alma 16:2.) Thus either the past-tense became or the infinitival become is theoretically possible here in 4 Nephi 1:43.
Ultimately, the real issue here in 4 Nephi 1:43 is whether it is textually possible to have the expression “begin to become”. Elsewhere in the text, there are no other examples of “begin to become” (although there are 111 occurrences of “begin to be”, with 105 of them with the past-tense form began, including here in 4 Nephi 1:43: “began to be proud”). Thus the use in the current text of “began to … become vain” in 4 Nephi 1:43 is unique. Semantically, “begin to become” is possible, but nonetheless it seems strange. Instead of this expression, the text relies on “begin to wax ” (7 times), as in 3 Nephi 2:3: “the people began to wax strong in wickedness and abominations”. Thus here in 4 Nephi 1:43, the reading of the printer’s manuscript (“began to be proud ... and became vain”) is probably the original reading.
Summary: Adopt the reading of the printer’s manuscript in 4 Nephi 1:43 (“began to be proud … and became vain”) since otherwise in the text there are no occurrences of “begin to become”.