Hey Melissa!
Welcome again to planDisney! I have dreams where I am eating Mickey-shaped Beignets with chocolate sauce while seated in the corner of
Scat Cat's Club - Café. I then cruelly wake up back home and nowhere near these delicious doughy bites of goodness. I'm excited that you are going to try and enjoy this with your grandma on your upcoming trip!
I reached out to a Cast Member and found that there is no specific boat or ferry that connects
Disney's Port Orleans Resort- French Quarter and
Disney's Port Orleans Resort- Riverside. There is a ferry that brings guests to Disney Springs and makes stops at each Resort. The Cast Member stated that the ferry usually does not operate early in the morning or late into the night so time out your travels for midday. He also stated that the schedule could change each day based on "operational needs." He advises you to speak to the Cast Members who work in boat transportation to confirm the timing and boat details.
As for drop-off locations, each Resort has its own dock. Disney's Port Orleans Resort- French Quarter is in the back of the resort on the other side of the Doubloon Lagoon pool area. You and Grandma will have to walk past the pool and down the center walkway to the main building. Disney's Port Orleans Resort- Riverside's boat dock is located behind the back of the main building where Boatwright's Dining Hall is located.
While I love the serenity of water transportation in Walt Disney World Resort, if your grandma is not doing well with her walking that day consider hailing an Uber/Lyft or cab to take you directly to the Resort you want to go to. The service will be able to quickly bring you where you want to go and leave you at the main building of each Resort.
I hope this helps and you grab some beignets to go!
Happy Planning!
-Nimmy