I would look under the pedal area of the platform to make sure the linkage didn't get disconnected, which is unlikely.
There is also a "Lever Kit" which moves the machine back and forth on top of the transmission which could be the issue, make sure to check that. Here is a link to the transmission page in the parts manual and it's item number 7 in the illustration.
John Deere Parts Catalog
There is also a drive belt kit and some other pieces to check which would prevent the machine from moving in either direction. You will see that there are few servicable items on the actual rear drive assembly and the transmission itself is priced at $372.14 and it is part number AUC11077 from Green Parts.
It's a common enough part that Amazon sells the entire rear transmission assembly, ready to install here
Amazon.com : General Transmissions Transaxle RT400 GT87137 : Garden Outdoor
Sadly, these machines are known for having a weak transmission and one which is prone to failure. The machines are real susceptible to failure if they are used to mow inclines or if the machine pulls anything of any weight or resistance.
The D105 use a transmission which is unique to those models which is made by General Transmission and is largely plastic gears and the unit is considered "Non Servicable",meaning the unit is supposedly factory sealed.
There are numerous videos and threads about working on these transmissions. Seems like there is a common gear failure, which if you can do the work yourself, makes the unit worth considering a rebuild. If you are going to need the dealer to repair the unit, make sure to get an estimate before you proceed as the labor costs along with the parts can sadly, make the cost to replace the transmission exceed the value of the machine for some.