This is not feasible
Atmospheric water must come from somewhere, and it cannot come from underground springs. The one and only option to do it in bulk is evaporation.
You don't describe your planet in detail, but let's assume that it's close enough to the sun to keep the poles melted. They won't be tropical (or the rest of the planet would be a blazing desert), let's assume they're arid.
That makes the rest of the planet fairly hot. Good. We need rapid evaporation. We have tons of trees and low tectonic activity so the water has worn most of the land fairly flat. (That's a problem by itself, BTW, because erosion is no small thing, but let's ignore that.)
You do get a ton of water into the atmosphere via transpiration (the plant form of sweating), but not nearly enough to create monsoons. For that you need open water to evaporate.
Could you do it with a swamp planet? Well... maybe, but if you're looking for no surface water, soggy swamp may not solve the problem. If soggy swamp could solve the problem, then you're looking at basically 100% humidity planet-wide. Yuck. Good for plants. Hard for critters (like humans) that need to breathe. (But, then again, sapient bipedal fish aren't unheard of.)
In reality, though, a planet with monsoons will have oceans, if only because an entirely flat planet (the only way you don't get oceans) is pretty unrealistic. Give a little altitude anywhere, even a few feet, and the water from those monsoons begins to puddle and gather. And you need them to evaporate enough water to create those tropical monsoons.
Think of Earth. 67% surface water, and we're NOT a tropical planet with regular monsoons. And yet you're looking to do it with 0% surface water. That's why I'm voting for not feasible.
But that's not the same as "don't use it in your story"
Having said that, IGNORE ME COMPLETELY. Oftimes authors get too bogged down in the details. The number of people who read your story and think to themselves, "that jungle planet with lots of rain and no oceans is sooo unrealistic!" compared to the number of people who think "This is the most awesome book in the world!" is so honking low that it's statistically zero. Up theirs (if you'll excuse my french). Have fun with your story and forget everything I said.