It is not clearly established what SCOTUS would rule, were such a case to arise, but it is highly likely that they would reject a requirement to have social media accounts as a requirement for holding a concealed weapons permit. The data under consideration in the question relate to the "good moral character" requirement of the new licensing law, which does not specify how good moral character will be proven. In reading Bruen, you obtain some indication of reasoning that the court would use is deciding the case. First,
to justify a firearm regulation the government must demonstrate that
the regulation is consistent with the Nation’s historical tradition of
firearm regulation
therefore courts must
assess whether modern firearms regulations are consistent with the
Second Amendment’s text and historical understanding
which means that
The burden then falls on respondents to show that New York’s
proper-cause requirement is consistent with this Nation’s historical
tradition of firearm regulation.
The new law is being contested in court, here is one example, and in the complaint, the malleable good moral character requirement ¶67ff is attacked, followed by attacks on the requirements said to be necessary to prove good moral character. In the US District court ruling, starting p. 90, the court analyzes the question of the "good moral character" requirement, finding that
the State Defendants attempt to avoid the impact of the
burden-shifting rule set forth in NYSRPA. They fail
...
the State Defendants do not clearly state why this regulation (or any
of the challenged regulations, for that matter) burdens a law-abiding
citizen's right to armed self-defense.
The state does provide historical precedent in their argument, pointing to colonial laws
forbidding the sale and trading of arms to Indigenous people,
prohibiting weapons possession by Catholics who refused to take
an oath of loyalty to the government, plus some state laws from 1776 and 1777
disarming persons based on their reputation for
being disloyal or hostile to the new Nation until they took an oath of loyalty – and a handful of city laws post 2nd Amendment pertaining to restricting possession of firearms from an individual who is potentially dangerous. But,
"the Court has
trouble finding them to be 'historical analogues' that are able to shed light on the public
understanding of the Second Amendment in 1791".
The decisions points to a distinction between restrictions on "readily apparent groups of people and often could be avoided by the objective
act of taking an oath". But
The CCIA’s "good moral character" requirement is not so objective in
nature (e.g., by requiring a finding of a likelihood of harm to self
or others based the prior conduct of the applicant, and permitting one
to avoid the restriction by taking an oath), and does not even
expressly recognize an exception for actions taken in self-defense.
As a result, based on a careful comparison of the burdensomeness of
the CCIA’s "good moral character” requirement (i.e., the burden
imposed in light of its justification) to the burdensomeness of the
relevant historical analogues (again, burden in light of
justification), the Court finds the burdensomeness of the CCIA’s "good
moral character" requirement (which is imposed on everyone and can be
avoided only through open-ended discretionary findings of
"temperament," "judgment" and "[]trust[]" by licensing officials) is
unreasonably disproportionate to the burdensomeness of the relevant
historical analogues (which were imposed on only readily apparent
groups of people and could often be avoided by the objective act of
taking an oath)
The court then enjoins the state from enforcing the provision requiring "good moral character",
the provision requiring the "names and contact information for
the applicant’s current spouse, or domestic partner, any other adults residing in the applicant's home, including any adult children of the
applicant, and whether or not there are minors residing, full time or part
time, in the applicant’s home", the provision requiring "a list of former and current social
media accounts of the applicant from the past three years", and
the provision contained in Section 1 of the CCIA requiring
"such other information required by review of the licensing application
that is reasonably necessary and related to the review of the licensing
application".
