It's a side effect of how the government was structured.

The legislative, judicial, and executive branches are intentionally separate for balance of power reasons.

The need for separation of power obviously applies to their core government functions. Now it might not be unreasonable to have the executive (being the "operations department" or J3 in mil parlance) take care of routine admin and logistics for the other two branches, simply for efficiency.

But that's not the way it was implemented... each of the three branches is almost totally self-contained in the sense that they don't rely on each other for care and feeding. For efficiency they are all supported by the GSA, but even that is a separate entity not falling under any of the three branches (it's funding and leadership appointments require both congress and the WH to concur).

The employer vaccine mandate is implemented as an executive order to OSHA, directing them to use their health and safety authority to mandate vaccinations. OSHA works for the executive, and has specific regulatory authorities. While it has authority over most workplaces, there are a few exceptions. For example the executive lets the DoD do their own osha stuff for obvious reasons. Same with FAA and airplanes. Since OSHA works for the executive it's authority inherently does not apply at all to the other two branches.

The federal mandate is just the customer defining the terms of the business arrangement, since the executive contracts for airline service (and most or all majors have those contracts). Apparently the terms of those contracts don't specifically exclude vaccine mandates, and the terms probably do have some sort of force majeure for health, safety, etc.

Bottom line, the executive cannot tell the other two branches how to run their organizations. The legislative and judicial could presumably implement their own internal vaccine mandates.

Also the ENTIRE federal government, all branches, could be subject to a vaccine mandate if congress passed a bill to that effect, POTUS signed it into law, and the courts didn't uphold any objections. But the executive can't do it alone.