I think at last count I've taken 32 Grand Slams, and just about every bird I've ever shot has been on public land, so YES...you most definitely can get it done that way! It's far cheaper to go that route, and you're not limited on time, or beholding to anyone else. Just plan your trip online, stick a few bucks in the wallet, and head on out. I camp wherever I hunt, and that in itself saves you tons of money. But, as has been mentioned, it can get mighty tough down there in the Sunshine State, when you're going for that Osceola. Their public lands get hammered pretty hard. Not saying it can't be done (I hunt there every year, and have killed 42 public land Osceolas), but you WILL have plenty of competition and/or hunter interference. If you're interested in a guided private land hunt, feel free to contact me, as I guide approximately a half-dozen folks annually, and we pretty-much always get our birds.
As for the Rio and Merriam's; public land access for both is MUCH easier. I've had great pubic land hunts for Rio's in OK, KS, CA, UT, NV, OR, and HI. If I had to recommend any over the others, it would be HI, followed by OK, KS, and UT. Merriam's opportunities are the easiest of all, with absolutely great chances in AZ, NM, CO, WY, SD, NE, MT, ID, and WA. I wouldn't be afraid to just pick any of these states and go.
If you just wanna pick my brain for additional info or hints, drop me a private message and and I'll see what I can do for ya. Good luck....there's nothing like getting your first slam, and especially so when you've earned every bird by doing it yourself on public land.