Religious accommodation is probably a red herring. Most state laws are weaker than federal law, and federal law - RFRA and RLUIPA - don't apply here.
Arguably, the false shipping notification is a violation of the FTC's regulations on mail, internet, and telephone order sales. See 16 CFR Part 435 at https://www.law.cornell.edu/cfr/text/16/part-435.
The regulation focuses on a default deadline of 30 days or a representation about shipping dates at the time of solicitation, so it may not apply; however, the FTC has taken action against companies that print shipping labels in advance but don't actually ship for some time. I started going down a rabbit hole, but had to cut it off, so I'm not sure what the right analysis on this piece. However, I'm not sure it matters, because...
What we really care about is wrongful termination for being a potential whistleblower. And you can be a whistleblower without being right. Retaliation for refusing to engage in an act the employee reasonably believes to be illegal is the real area of legal risk here. However, until/unless the FTC Whistleblower Act becomes law, https://www.congress.gov/bill/117th-congress/house-bill/6093..., this is a question of state law.
TL;DR: Classic lawyer answer: "it depends." If the state they're in protects employees who object to conduct they reasonably believe to be illegal, and the employee reasonably believes the false statements about shipping are unfair and deceptive trade practices, that's probably the ballgame. If the state they're in doesn't offer that protection, then maybe not.