I've never seen software that does what you're asking for JPGs, only RAWs. The main issue is that any profile or adjustment that is computed from a JPG photograph of a particular chart would only apply to other JPG photographs with the same white-balance.
JPGs don't use "camera profiles" anyway in Lightroom, the profile is always embedded.
There was a product that used to compute HSL slider offset values given a JPG photo of a their own proprietary chart that looked similar to a color-checker but had two of the rows reversed so the software couldn't be used with a standard color-checker, unfortunately. Now their chart has twice as many (48) patches and looking on their website I'm not as sure it still does work with JPGs or if it just does raw files, now.
The chart and software are called SpyderCheckr from Datacolor. There are various packages that do different things but none use a ColorChecker, so having one is mostly a waste until you get a camera that can shoot raw photos, so keep it in a humidity and temperature controlled environment as much as possible until you have the right camera. You can probably use the gray patches to adjust Exposure but that's about it.
Nowadays you don't need a DSLR to have raw photos, though. My Galaxy S6 phone with the latest Marshmallow update and camera software update will take raw photos in DNG format in Pro mode which I can create custom camera profiles for using a ColorChecker, although the camera is really good by itself, so when I first found it could do DNGs I took quite a few, but now, I usually just use the camera to make JPGs unless the shot is really important and the lighting isn't too harsh. I think the S7 phone camera is even better.
What kind of camera do you have? Sometimes there are firmware hacks that allow the camera to shoot raw format even if the standard camera can't.