So I've not been, but I've heard talk about it too. Having a read on Wikivoyage's article certainly seems like it's possible sans-tour.
There are daily flights from other Brazilian cities such as Campo
Grande and Cuiabá to the gateway city of Corumbá, from where tours can
be organised. It is possible to get into Pantanal from Bolivian side.
You can take the train from the Bolivian city of Santa Cruz to Puerto
Suarez, cross the border here and use taxi or bus to reach the city of
Corumbá - from cities of Puerto Suarez and Puerto Qujarro there are no
organised visits due to nature protection, politics or incapability of
From Campo Grande you can catch a bus to go into the Pantanal.
Depending on where you plan to go it may take between 4 to 5 hours to
get into the Pantanal. In the bus station in Campo Grande catch the
bus to Corumba. A good place to get of is Buraco das
Now I'm unsure how much solo-travel you've done, but often it's way easier to find information once you're there, and it becomes much easier. It's important to realise that if people live in the area, then there is always a way to get around that doesn't require a tour.
On the ground, make sure to speak to every traveller you meet coming the other way, and speak to locals. Ask about road conditions, buses they recommend, and more.
Once there, it depends what you want to see. If it's wildlife, as per the article, you're pretty much guaranteed to see it. If it's certain locations, then you'll have to ask people about how to get there. Ask a local to take you. Get to know tour companies and ask if there's a cheap way - often they'll recommend it - after all, if they love their job, it's because they want people to see it, not just make money - so if you can't take their tour they may suggest alternative options.
Good luck, and any info you do find, you can add as an answer when you get back :)