Can I get a bank loan or mortgage against my share in a non-profit housing co-op in Ontario?

Generally not in the way you could mortgage a house or condo. A mortgage is a security interest registered against real property title, and a co-op membership share isn't real property — it's an interest in a corporation, paired with an occupancy agreement, with no title for a lender to register against. Because of this, conventional mortgage financing doesn't really apply to a co-op membership the way it applies to a freehold or condo purchase.

In practice, this means the membership share or deposit required to join a co-op is typically paid directly by the incoming member, rather than financed through a registered real property mortgage the way a down payment and purchase price would be for a house. Some members may still be able to borrow the funds through other means, such as a personal loan or line of credit unrelated to the co-op unit itself, but that's a personal financing arrangement rather than a mortgage tied to the co-op unit. Anyone planning to join a co-op and expecting to finance the membership share the way they would finance a home purchase should confirm early what payment is actually required and how it can be funded, since assuming mortgage-style financing is available can derail the application process.