Could you consider, if you haven't already, having a feature where transporters can be rated/reviewed. Maybe similar to Yelp or there is a software development community, is it GitHub?, that allows for reviews and ratings?
Making it easy to filter down and contact the most viable volunteers, by location and day/time availability. A mapping feature would be useful, if you could plug in the starting location and the destination and filter to resources that would be along the route. I had built something similar using Google Maps for a small rescue that covered a tri-state area. The map pins were interactive and could allow us to identify and contact specific volunteers.
Have you looked into the design and features that are used by Uber/Lyft? I know there are some other companies that have built service platforms similar in other industries, like uberized healthcare services - I forget the company's name, but they provided nurses and phlebotomists for patients that were at risk or had difficulty getting to a medical providers office. They had a call center and dispatch set up.
I love your idea and as a market strategist, would love to help more if I can. If you'd like to set up a time to discuss anything, please feel free to send me a direct message.