Vegetable oils are triglycerides with mostly 18 carbon chain acids (oleic, linoleic acid), some 12, some 16. Keep in mind octane is 8 and has no oxygens for hydrogen bonding. You have to process it like biodiesel to get glycerol and
https://en.wikipedia.org/wiki/Fatty_acid_methyl_ester to get something even remotely combustible and the boiling point is still extremely high. I don't know how easy it is to make biodiesel at home with methanol and alkali.
Glycerol I think actually does improve octane and can be diluted in ethanol, but you can't use much of it since it's very heavy. It's basically a bunch of alcohols chained together. The esters are AFAIK very low octane but maybe if you mix some heavy aromatic solvents as octane boosters it might work with high intake air temperature.
Spark ignition engines really need low molecular weight hydrocarbons that evaporate easily, that's the big limitation. Maybe if you could separate the esters to burn in diesel engines then burn the trace methanol and glycerol in a gasoline engine mixed into a lot of gasoline it would work, but I don't know how that would be done at home.
https://pubs.acs.org/doi/10.1021/acsenergylett.6b00421