Not sure on the specifics of paperwork regarding the actual lease and payments, but:
This is essentially going to be a used vehicle purchase. It is the seller's responsibility to provide you with a Used Vehicle Information Package (UVIP) which they can purchase from ServiceOntario for ~$20 iirc. You will need some paperwork contained in the UVIP to complete the title transfer. The UVIP will contain information regarding the history of the vehicle (the same kind of things you'd find in CarFax, service history, any prior accidents, etc).
Make sure you get the vehicle inspected at a reputable mechanic prior to purchase. You and the owner of the vehicle will need to come to an agreement on who pays for any repairs needed.
You will also need to pay sales tax on the purchase price of the vehicle when you do the title (Not sure how this works, because the lease payments you'll be making will also include the sales tax on the original purchase of the vehicle). See if you can convince the owner to just sell you the car for the remaining balance of the lease (minus tax if they'll agree to that). This way they can pay off the remainder of their loan, saving you both the interest on these remaining payments.
For example: My current lease costs me $60/month in interest alone. If I were to pay for 12 months right now as a lump sum, I'd actually be making a net $720 gain by avoiding $60*12 in interest.