I know there are no take off requirements for part 91. But if we decide to use an ODP, legally should our airplane be able to make minimum climb rate of takeoff minimums described in the ODP chart?
Legally, the only time takeoff minimums apply in part 91 is if you are assigned and accept a clearance that includes them. ATC sometimes assigns an ODP, and you must reject the clearance if you cannot meet the minimums.
If it is not part of your clearance, you do not have to meet the takeoff minimums from a legal standpoint. But you are on your own for obstacle avoidance at that point- the obstacles aren't any lower because you're flying under part 91.
The FAA could still take action under 91.13 if it considers you flying the ODP to be reckless. And of course flying part 91 is little comfort if you fly into the side of a mountain.