I've tried them my self and almost all canadian banks allow Online banking.
For the first account you can apply online then print and sign the suscription form and send it to the bank you applied for, with a $25 Personal Check (The check must have your name and adress on it).
This part can be made faster if you go to a branch.
Once the first account is opened, you can open other saving and checking accounts online. Most of the times the account will be ready instantly or within the next buisness day.
Here are the banks I'm talking about :
R B C..... T D Canada T rust ..... S co tia Bank .... IN G direct Canada.
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