The relevant passage from the Church Handbook of Instructions (pg. 30) says this:

"One or more Melchizedek Priesthood holders must consecrate olive oil before it is used to anoint the sick or afflicted. No other oil may be used."

[skip a bit]

"If consecrated oil is not available, a blessing may nevertheless be given by the authority of the priesthood."

My comments:

I think that says it pretty clearly: No other oil! Not other vegetable oil, not crankcase oil.