Office of Deputy Mayor isn't even mentioned in the City of Toronto Act, which means it's purely a council construct.
The Municipal Act also applies (subject to any modifications to it made by the City of Toronto Act). It is well accepted law that the general powers conferred by the Municipal Act*** enable municipal councils to provide for the appointment of a DM by council and that the City of Toronto Act enables Toronto city council to delegate (or not) that appointment power to the Mayor.
***The Municipal Act confers a number of subject-matter specific powers on councils and places a number of subject-matter specific limits or conditions on the powers of councils. But, subject to any specific rules to the contrary, the Municipal Act also confers on councils general powers. So the issue comes down to whether providing for the appointment of a DM (as some municipalities do and some do not) comes within those general powers; the answer to that is yes.
MUNICIPAL ACT
PART II
GENERAL MUNICIPAL POWERS
Scope of powers
8. (1) The powers of a municipality under this or any other Act shall be interpreted broadly so as to confer
broad authority on the municipality to enable the municipality to govern its affairs as it considers appropriate and to enhance the municipality’s ability to respond to municipal issues. 2006, c. 32, Sched. A, s. 8. [
emphasis added]
Ambiguity
(2) In the event of ambiguity in whether or not a municipality has the authority under this or any other Act to pass a by-law or to take any other action, the ambiguity shall be resolved so as to include, rather than exclude, powers the municipality had on the day before this Act came into force. 2006, c. 32, Sched. A, s. 8.
...
Broad authority, single-tier municipalities
10. (1) A single-tier municipality may provide any service or thing that the municipality considers necessary or desirable for the public. 2006, c. 32, Sched. A, s. 8.
By-laws
(2) A single-tier municipality may pass by-laws respecting the following matters:
1.
Governance structure of the municipality and its local boards. [
emphasis added]
2. Accountability and transparency of the municipality and its operations and of its local boards and their operations.
3. Financial management of the municipality and its local boards.
4. Public assets of the municipality acquired for the purpose of exercising its authority under this or any other Act.
5. Economic, social and environmental well-being of the municipality.
6. Health, safety and well-being of persons.
7. Services and things that the municipality is authorized to provide under subsection (1).
8. Protection of persons and property, including consumer protection.
9. Animals.
10. Structures, including fences and signs.
11. Business licensing. 2006, c. 32, Sched. A, s. 8.