It would be nearly impossible to do so legally.
The monarchy is a constitutional issue and has a distinct legal persona in each of the provinces and the federal government. In order to remove it the House of Commons, the Senate, and the legislative assembly of all 10 provinces have to consent.
This poses 3 major challenges to republicanism.
First of all, the House of Commons would have to be in agreement to abolish the monarchy. The Conservative party, under Harper, is pro-monarchy, and the Liberals' most recent party vote on the issue came in favour of the monarchy (or at least against abolishing it). If the NDP were to try to abolish the monarchy they would need support from the other parties, as they aren't likely to gain a majority. The only party that openly supports a republic is the Bloc Québécois, and allying with them will alienate the NDP.
Secondly, the Senate has to approve. Right now, the Senate has a Conservative majority, therefore isn't likely to support republicanism. There are 22 seats not filled, so theoretically the tide could be changed if the Independents were swayed to republicanism, however given Mulcair's strong anti-senate stance he'd lose some political clout if he filled those seats.
I believe that is not exactly correct. A constitutional amendment to abolish the monarchy in Canada would fall under Section 41 of the Constitution Act, 1982 (quoted below), which, as you said, requires the approval of the House of Commons, the Senate, and the Legislative Assemblies of all 10 provinces.
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Section 41 [Highly Qualified Proceedings][FONT=Arial,Helvetica]
An amendment to the Constitution of Canada in relation to the following matters may be made by proclamation issued by the Governor General under the Great Seal of Canada only where authorized by resolutions of the Senate and House of Commons and of the legislative assemblies of each province:
(a) the office of the Queen, the Governor General, and the Lieutenant Governor of a province;
(b) the right of a province to a number of members in the House of Commons not less than the number of Senators by which the province is entitled to be represented at the time this part comes into force;
(c) subject to Section 43, the use of the English or the French language;
(d) the composition of the Supreme Court of Canada; and
(e) an amendment to this part.[/FONT]
However, Section 47 of the same Constitution Act, 1982 (also quoted below) includes an exception under which a constitutional amendment may be made under Section 41 above (and a few other sections) if the Senate fails to approve it within 180 days (not counting days when Parliament is prorogued or dissolved), and the House of Commons subsequently passes it a second time. In other words, the Senate could only delay the abolition of the monarchy, but it could not prevent it indefinitely provided that the House of Commons and the Legislative Assemblies of the provinces agree to turn Canada into a republic.
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[FONT=Arial,Helvetica][FONT=Arial,Helvetica][FONT=Arial,Helvetica]Section 47 [Default Adoption]
[/FONT][FONT=Arial,Helvetica][FONT=Arial,Helvetica](1) An amendment to the Constitution of Canada made by proclamation under Sections 38, 41, 42, and 43 may be made without a resolution of the Senate authorizing the issue of the proclamation if, within one hundred and eighty days after the adoption by the House of Commons of a resolution authorizing its issue, the Senate has not adopted such a resolution and if, at any time after the expiration of that period, the House of Commons again adopts the resolution.
(2) Any period when Parliament is prorogued or dissolved shall not be counted in computing the one hundred and eighty day period referred to in Subsection (1).
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