I've used airportshuttle in the past, but they no longer accept individual riders and only book groups of 10 or more. RTA, the transit authority for the area, also transports me as I hold an ADA paratransit card for my own local agency. Allow yourself about 3-4 weeks before your trip to notify them if trying that as they tried, without luck, to verify my info beyond my card. They were still quite helpful and allowed me access to their shared ride services when my agency failed to respond promptly.
In New Orleans proper, the RTA busses and most streetcars are wheelchair assessable, with the exception of the St. Charles streetcar line as they only have historical equipment.