We never add an oil to an injection and I can't imagine why you would. There is enough fat in most BBQ meats that there is certainly no need for oil. I would also worry about the taste and texture if oil were added. No oil for me!
Jambo J-3, Jambo Backyard, Backwoods Party, Backwoods Chubby, Custom Offset, WSM, and a Primo Oval XL.