You didn't mention the departure time, which train in Lyon you want to catch or which Avignon station you'd like to depart from... but generally, for French TGVs you can book a 1st class reservation with a pass for 2nd class. It's just a bit more expensive. You can book that:
Keep in mind that the Interrail website currently does not show the 20€ TGV reservations that are available after the 10€ ones have sold out. Belgian railways (see link above) will also show the 20€ seats, which are available until the train is full.
Also keep in mind that there are 2 stations in Avignon. Avignon Centre is close to the city centre but Avignon TGV is just south of the city and it will take time to get there (there are trains).