This question is aimed at understanding the viewpoint of those groups who endorse the doctrine of biblical inerrancy as outlined in the Chicago Statement. I understand that there are likely sub-groups under this umbrella, and this is intended as an overview question of perspectives within that group. From the statement:
WE AFFIRM that inspiration, strictly speaking, applies only to the autographic text of Scripture, which in the providence of God can be ascertained from available manuscripts with great accuracy.
Clearly, neither the consonantal decisions of the Masoretic text nor vowels of any sort were part of the autographs, with the implication that these are, strictly speaking, outside the purview of the statement. However, as noted above and in the linked answer, people (myself included!) seem to feel uneasy when encountering emendations to the Masoretic text.
To what extent do various traditions under the umbrella of 'inerrancy' value the Masoretic text, both its vocalization and consonantal textual decisions? This may include the degree to which these decisions are considered inspired, instructive, illustrative of authorial intent, or whatever other facets of 'valuable' can be brought to bear.
A few examples of questions on Biblical Hermeneutics where we have discussed emendations to the MT that are understood by many Christian translations: Psalm 22:16, Deut 32:8, Psalm 19:4, Obad 7, Ecc 7:27, Job 6:14, Hosea 11:12.
These include some examples where the emendations are purely conjectural and others where they are based on alternative witnesses, most prominently the Dead Sea Scrolls and the Septuagint.