It all depends upon how far you want to drive.
I lived in Carmel for 16 years. Born and raised near Noblesville, and currently live between Anderson, Pendleton, and Lapel. This is my home turf.
Around Carmel, you will not find any nearby campgrounds. The closest will be White River. (If you are able to get a spot. They are constantly filled.)
There are a number other campgrounds around, but will require a little driving.
My next suggestion would be Mounds State Park in Anderson. Traditionally, beginning on April 1, water is turned back on in all state parks in Indiana. But, if the weather is still freezing, they could delay. Anderson is a nice state park. You'll have electric, fill your water tank at a fill faucet (there are several now), and dump at the dump station. We live about 12 miles from Mounds in Anderson and camp there at least once a year over a week end. Indiana State Parks are open all year, but water is turned off from Nov 1 to April 1.
Next state park is Summit Lake, but this is a much farther drive. But we really enjoy Summit Lake.
Then, there is the state park camping at the Indianapolis State Fair Grounds. But, this is really metropolitan type camping, inside the Fair Grounds. But you do have full hook-ups. (I would never go there myself).
Next are the few private campgrounds in the area.
Glo-Wood, in the country, nothing there but corn fields and a place to park, but full hook-ups. (and only 5 miles from where I live) Click here
They have a lot of permanent campers there. But also have a transient section too. Not bad for bedroom parking though.
There is Pine Lake on I-69 at the Pendleton Exit. I pass this one almost every day. They have cabins and day-fishing. Web information says 100 RV sites, but I've never seen any RV's there, unless they are on the back side of the lake and cannot be seen from the road. Click here
There is Hartland RV Resort on I-70. I've never been to this one: click here
S&H Campground is not to far from Hartland, but a little farther to drive to Carmel. If nothing else works, you might try there: Click here
Raceway Family Campground is on the West side of Indianapolis, not too far from the airport. Never been there, so cannot make any recommendations. It will be drive around the North side of I-465, hit US 31 North and go straight to Carmel. (Probably about 1 hour drive) Click here
Old Mill Run is a good hour drive to Carmel. My mother in-law lives about 1 mile from there. Take I-65 South to Highway 32. Follow 31 to US 31 South, go to Carmel. Click here
A little further drive, near New Castle, Indiana is Cornerstone campground. We've been there several times, and I recommend this one as one of the better private campgrounds around. It is a farther drive to Carmel though, probably 1 hour 15 minutes or so. Click here
There are two KOA's in the area. One is along I-70 going East toward Ohio. It is right along the Interstate and you get all that Interstate traffic. Click here
The other KOA on the North East side of Anderson, off Highway 32 heading toward Chesterfield. This is also a very nice campground .... BUT....
The when the campground was still run by one man and he owned it, the owner was a real ***. And I'm not exaggerating. They had several long term campers there. Transient campers were treated like criminals.
He franchised with KOA a couple or 3 years ago and he still runs the campground. We've never been back because he still controls the campground and last I heard, he's still a tyrant. It's called Anderson/Muncie KOA, but it's really on the North East side of Anderson: Click here
These are all your available options.
There was one little campground just East of Carmel and I've driven through it. The name of it was NotNothingMuchHere. Really, that was the name of it. I've not been able to find it on the Internet for a few years now, so I don't know if it still exists. But trust me .... there really was Not Nothing Much there! Just a place to park the camper and have electricity and water. The name fit the campground perfect. I think it was a bed-room campground for the Indy 500.
And oh, Sleepy Bear by the music center at Hamilton Town Center, I've not seen any campers there at all, even for the music events in a few years now. But tent camping in different people's back yards has sprung up all over around the music center. Signs go up when the big events come, and signs go down when its over. The music center has a gigantic tenting campground for events. But when the event is over and people are gone, that campground looks like a bomb exploded. Trash everywhere! People are absolute pigs. It takes them a week to get all the trash picked up off the ground, broken left quick shades, and tents people leave behind because they don't hold up to these strong Indiana winds and torrential rains we get.
I know this is a long post, but I hope this helps.