The relevant portion of 1 Peter 2:8 is:
... οἳ προσκόπτουσιν τῷ λόγῳ ἀπειθοῦντες εἰς ὃ καὶ ἐτέθησαν.1
The differences arise depending on (a) what you do with the nominative masculine plural οἳ at the beginning of the clause quoted above and (b) how you understand the relationship between the verb and the participle.
οἳ could be either (1) an article or (2) a relative pronoun:
Article: If οἳ is taken as an article and associated with the participle ἀπειθοῦντες ("disobeying"), the complete subject would be "the people disobeying the word." This would be translated to the effect: "The [people who are] disobeying the word stumble...."
Relative pronoun: Achtemeier argued that
it is more likely that [οἳ] functions as a relative pronoun, referring to the ἀπιστοῦσιν [("unbelieving")] of v. 7, with the ἀπειθοῦντες functioning as a circumstantial participle of cause (“They, namely, the unfaithful, are the ones who stumble because they disobeyed”). Again, while one could construe the τῷ λόγῳ with προσκόπτουσιν (“they stumbled against the word”), the use of ἀπειθέω (“disobey”) with τῷ λόγῳ (“the word”) in 3:1 and with τῷ τοῦ θεοῦ εὐαγγελίῳ (“the good news of God”) in 4:17, both in reference to unbelievers, makes it likely it ought to be construed the same way here (“They stumble because they disobey the word”).2
It could still be translated other ways depending on how you view the relationship of the participle to the main verb, such as:
- "... who stumble through disobeying the word...."
- "They stumble as they disobey the word...."
Those who use "being disobedient" (or "disobeying") are likely attempting to translate the participle literally without interpreting the relationship between the participle and the verb.
You also have to make a decision about the subject of this clause regardless of how you understand the function of οἳ, which must be determined based on context.
1 Barbara Aland et al., eds., The Greek New Testament, Fifth Revised Edition (Stuttgart, Germany: Deutsche Bibelgesellschaft, 2014), 1 Pe 2:8.
2 Paul J. Achtemeier, 1 Peter: A Commentary on First Peter, ed. Eldon Jay Epp, Hermeneia—a Critical and Historical Commentary on the Bible (Minneapolis, MN: Fortress Press, 1996), 162.