Google Flights doesn't even know LGQ, so it's quite likely there is no public service to that airport at all. Flightradar24 doesn't recognise it either. Flightaware lists a few days with a few flights back in March and nothing since then.
Petroecuador is the national oil company of Ecuador, so they probably just operate their own private flights there.
Google Maps shows a photo from March 2023 of an Aeroregional aircraft in LGQ, but it's not listed as a destination on their website at this time.
There is however another airport in El Coca (OCC), less than 2 hours away by road.
But Lago Agrio / Nueva Loja airport is less than 6 hours away from Quito by road, so it may be easier that way, especially since you need a vehicle to get to the reserve anyway.
There's actually another airport even closer (Tarapoa Airport, TPC), but it does not seem to have any public service either (probably even less than LGQ).
Note that while I understand how you get near the reserve, I have no idea how you get into it or to any of the lodges within. My impression is you get there by the river rather than by road. It would probably be a good idea to contact the lodge you want to stay at, they probably have better information on how to get there, and it's possible they arrange private flights to the most convenient airport, or arrange transportation from Quito via other means.
