Yes Discount Tire will install and mount both used and new tires even when bought from somewhere else.
However the mounting and balancing cost of the used or new tires that are bought elsewhere typically costs around $30.00 per tire.
So that cheap used tire that you bought may not sound so cheap after all.
If you buy the tires from Discount Tire they will mount and balance them for around $10.00 to $15.00 per tire.
So it's best to buy the tires from Discount Tire but if you bought some used tires from someone and need them mounted and balanced then Discount Tire will do the mounting and balancing of the tires but at a cost.