I am not sure if you object to palm oil, but I think it is probably the best substitute for animal fats, to give you the amount of hardness you need. Most vegetable oils are very soft oils. Coconut oil contributes to hardness, but too much of it makes for a pretty harsh soap.
I was going to suggest that you add a little stearic acid for hardness, but I am not sure if it is derived from animal fats.
Another option--though expensive--would be to use quite a bit of cocoa butter. Mango butter and shea butter will also contribute to hardness. Soaps with a high percentage of butter are quite luxurious.
Yet another option is to make a Castile or Bastile soap--that is, either pure olive oil or about 80% olive oil. But such soaps are not long-lasting without a LONG cure time, and they don't lather well without a long cure time. They cure to very hard soaps, though, and can be made to be very hard very quickly with a steep water discount.
The first batch of soap I ever made (about six years ago) was 100% olive oil with oatmeal and honey. I still have three bars of it. During my first year or two of making soap, my go-to recipe was a Bastile soap that was about 80% olive oil and 20% almond oil. A steep water discount--and a fairly long cure--is advisable with a recipe like this, too.
After along cure, such soaps are just marvelous and lather really well.
But your best bet is to use SoapCalc to develop a recipe with the desired qualities. http://soapcalc.net/calc/SoapCalcWP.asp