The answer
No. Receiving addresses come from the configured multisig wallet data, so participating hardware signers need not be online for deposits.
What to do next
Use a watch-only coordinator and verify the address when initially setting up.
Trezor hardware wallets
Potential cold-storage product; not a substitute for safe backup procedures. Availability, supported assets and local eligibility must be checked before signing up or buying.
Visit Trezor hardware wallets ↗Official provider link · Compare suitability and availability before choosing.