Canada Post is a corporation created by an Act of Parliament, namely the Canada Post Corporation Act, RSC 1985, c C-10 ("CPC Act"). Although Canada Post is structured as a corporation, it is required to "maintain a corporate identity program ... that reflects the role of the Corporation as an institution of the Government of Canada". CPC Act § 5(2)(e). This seemingly suggests that the legislature intends for Canada Post to be thought of as part of the government. The corporation fulfills a role specifically assigned to the federal government in the constitution, namely the provision of "Postal Service". Constitution Act, 1867, 30 & 31 Vict, c 3, § 91(5). The corporation also enjoys a number of privileges not normally accorded to private corporations, such as a monopoly on certain services (CPC Act § 14) and a general immunity from civil liability (CPC Act § 40(1)).
I think it is likely that Canada Post is part of the "government" within the meaning of § 32(1)(a) of the Canadian Charter of Rights of Freedoms (Part I of the Constitution Act, 1982, Schedule B to the Canada Act 1982 (UK), 1982, c 11) ("Charter"). This means that it is unlawful for Canada Post's operations to constitute substantive discrimination on the basis of, among other things, age or disability. Charter § 15(1). When making discretionary decisions, government agencies must explicitly "balance[] the Charter values with the statutory objectives". Doré v. Barreau du Québec, 2012 SCC 12 at ¶ 55. Failure to do so in a reasonable manner will result in the actions of the agency being set aside on judicial review. Id at ¶ 7.
Based on the above principles, anybody who is adversely affected by the phasing out of door-to-door delivery is free to file an application in a "court of competent jurisdiction" to obtain "such remedy as the court considers appropriate and just". Charter § 24(1). I do not know whether anybody is yet challenging this phase out, but based on the arguments above in the thread, it sounds like it would be an interesting case to litigate.