I know you said 30 minutes but there is nothing great within that radius. Raceway Campgrounds as mentioned above is a huge empty grass field next to the drag racing track west of town.
If it was me, I would make a long weekend of it and camp out at Shades SP. It would be about an hour downtown but is absolutely a great place. You could still do your day trips to Indy but also enjoy a great trail system and nice campground as well.
Johnson County Park has camping, with water and electrical hook ups. They also have a new shower house and laundry. The are about 25 miles south of Indy just south of Franklin Indiana....(between Franklin and Trafalgar off SR 252)
Read all about it here: JOCO Park Campgrounds | Johnson County Parks and Recreation
Good luck
You might try White River Campground in Hamilton Co. Its near Noblesville (Cicero technically) which is north of Indy. It's next to highway 37, probably a little farther than 30 minutes. Hamilton County, Indiana / White River Campground Nice place, it's also right next to Kotweei Park, (no camping) Hamilton County, Indiana / Strawtown Koteewi "Prairie" Park ....had to make a first post sometime, hope those links work!
There is a KOA right off 70 @ mount comfort road... its outside marion county and away from that typical riff wraff.... Don't know about facilities though I have never been down the drive..... Just down the road from that is another privately owned place that you always see lots of decent appearing families around it... might be something to look into.
You might try White River Campground in Hamilton Co. Its near Noblesville (Cicero technically) which is north of Indy. It's next to highway 37, probably a little farther than 30 minutes. Hamilton County, Indiana / White River Campground Nice place, it's also right next to Kotweei Park, (no camping) Hamilton County, Indiana / Strawtown Koteewi "Prairie" Park ....had to make a first post sometime, hope those links work!