应该是可以的吧。但是有条件的。。
首先，要有SIN号码才能工作，一般 study permit 印有以下内容才可以申请到SIN number：

May work 20 hours per week off campus or full-time during regular breaks if meeting criteria outlined in paragraph 186(v) of the Immigration and Refugee Protection Regulations.0

May accept employment on or off campus if meeting eligibility criteria, per paragraph R186(f), (v) or (w) and must cease working if no longer meeting these criteria.

然后再看看上面提到的 R186(f), (v) or (w) 的具体内容是什么：

186 A foreign national may work in Canada without a work permit

(f) if they are a full-time student, on the campus of the university or college at which they are a full-time student, for the period for which they hold a study permit to study at that university or college;

(v) if they are the holder of a study permit and
(i) they are a full-time student enrolled at a designated learning institution as defined in section 211.1,
(ii) the program in which they are enrolled is a post-secondary academic, vocational or professional training program, or a vocational training program at the secondary level offered in Quebec, in each case, of a duration of six months or more that leads to a degree, diploma or certificate, and
(iii) although they are permitted to engage in full-time work during a regularly scheduled break between academic sessions, they work no more than 20 hours per week during a regular academic session;

(w) if they are or were the holder of a study permit who has completed their program of study and
(i) they met the requirements set out in paragraph (v), and
(ii) they applied for a work permit before the expiry of that study permit and a decision has not yet been made in respect of their application;

然后对照你自身的情况看能不能满足条件。你的情况应该属于(w)吧，在学签到期前递交工签申请应该是可以工作的。

参考网站：
https://www.canada.ca/en/immigration-refugees-citizenship/services/study-canada/work/work-off-campus.html
https://laws-lois.justice.gc.ca/eng/regulations/SOR-2002-227/section-186.html