First, is your Blazer under factory warranty? You haven't given details on mileage. Obviously, if you're under warranty still, then I would imagine that could be considered a factory defect and fixed under warranty. If your Blazer is not within the factory warranty, you'll need to contact the dealer and ask them if just fixing the button is doable and then order parts and have at it. Either way, you'll need to contact a Chevy/GM dealer and ask them if you can just replace the button.
I would also pull up the diagram on your steering wheel online and look at it (Google). You should be able to see an exploded version of it with all the pieces listed around the wheel... maybe. This will give you a better visual of what it takes to get inside that area, that's even if you can. The steering wheel might simply come as one unit, most likely, and then you're looking at buying a steering wheel. I went ahead and pulled up some diagrams online of these wheels and it looks like after you pop out the centerpiece, airbag/horn, you might be able to access the buttons on each side. It looks like they can be popped out, maybe as a cluster assembly. You'll need to put your eyes on that and confirm, though. There are many diagrams on this wheel out there, so you should be able to see a wealth of information on it. Just Google 20** Blazer Steering Wheel.
Note: I found this link that shows you can replace just the cluster assembly. I'm not sure if you're tackling this yourself or having a dealer do the work for you. In the event you do it yourself, here is a link that shows a diagram of a wheel. Not your wheel necessarily, but similar. The dealer might be able to email you a diagram of your wheel. Ask them.
Genuine OEM GM Part - Cruise Control Switch (Left) 13547746 - GM Parts Overstock | GM Parts Overstock
I replaced a steering wheel on a 2016 Maxima I purchased new. The steering wheel had a cut in the leather wrap when I purchased it and the dealer wouldn't cover that under that factory warranty for me. I wound up purchasing a new steering wheel, but I went back to the parts dept. and I got them to sell me a new one at close to wholesale pricing. The crappy thing about that though is at even wholesale pricing, it was still around $900.00. I never had to dig into the buttons for any reason, but I just wanted to share the cost of that with you so you don't get sticker shock if you have to purchase the whole darn thing.
Definitely contact the dealer and find out the path to that fix. I hope you can fix that with an easy fix. You may know this already, but if you get to fix it yourself, once you've pulled the wheel off, you have an airbag connector behind the wheel. Be very careful. Disconnect the battery first before pulling the wheel off the center post. Once the wheel is off, note the grooved ring around the post. You'll want to be sure to centerline your wheel going back on with those groves. There should be a mark from the factory indicating alignment, so it's pretty cut and dry. I know when the dealer put my wheel back on for me, they were 2 notches off and the wheel was a little cocked from the center. Had to take it back so they could correct that.
Good luck, and may the force be with you!