IANAL, but: as an employee, you are working for a company that owns your work, a manager that manages you, as well as any software/IP you produce, the PC and network you use, etc. So they can require a copy of your login/password to the network.
In a way, the manager and the company as a whole doesn't need a copy of your password; I'm sure the network administrator has root access to all accounts on the network, can access the passwords, and can change them at will to give or prevent your access.
By asking to "write the password to a piece of paper, place it inside an envelope, and seal and sign that envelope," the manager is reasonable in
needing "access to the account when she is unreachable," i.e. out for the week, during a network security situation, or after you don't work there anymore.
They may also want some sort of paper trail of your original password rather than depend on the network. That motivation is their own management style, possibly in conjunction with the network administrator, and/or some requirement thought necessary by the legal department.
If you decline, they may tell you the network administrator has your password anyway. But, to decline such a request as a new employee is not a good idea.