I want to create an inland forest on the western side of a large continent that features cenotes instead of rivers. Is this possible, and if so, how can I make it work? Any requirements for altitude or soil structure are acceptable, so long as the area can support a fairly dense forest biome.
This is possible and we actually have an example on Earth (at Yucatan peninsula).
"Mexico's Yucatan peninsula is low and relatively flat with no surface rivers or streams. However, below the ground run the three longest underground water systems in the world (Ox Bel Ha 180 km; Sac Aktun 172 km; Dos Ojos 82 km) which have directed the distribution of human settlement on the peninsula for the last 10,000 years...the world of the cenotes and underground rivers. (...) Cenotes were the only source of water in the jungle for the Mayan civilization and are considered sacred by the Mayan people."
What you want from geologic point of view is flat land and soluble rocks below soil (for example limestone). For exact details it would be the best to refer to multiple sources available on Yucatan geology.