Your camera is supported by Lightroom 6.8 and higher.
I have lightroom 4 now and it won't import from my Sony A6500.
I WILL NOT GO TO THE NEW ONLINE ONLY VERSION!.
If I purchase LR 6, and update to the latest out there, will it import the files from my sony A6500?
Lightroom Classic CC (also known as Lightroom 7) is not a ONLINE ONLY version. It runs entirely on your computer, all of your photos are on your computer (or attached disk), and you never have to upload a single photo to the internet.
Lightroom 6.14 will never be updated -- no bug fixes, no new features, no new camera support, no new lens support, and will not run if an operating system change causes it to stop running. Also, no technical support or phone support from Adobe.
As a point of clarification, Lightroom Classic CC is not "online only". It is subscription only, which means that it has all the latest features and is updated periodically with support for new cameras, new features, etc. However, it still is installed on your computer, downloads and imports images to your local hard drive and functions just as your current version of Lightroom functions. Yes, there is a subscription cost to it. And if that is what you are resisting then Lightroom 6 is available. But there are no more updates available for it EVER. Once you have installed that program, and installed the 6.14 update, that is all you will ever have. If that is what you want then it is available.
so to be clear, if I get LR 6 and update it, it will import my Sony A6500 photos?
Yes, you need to have at least Lightroom 6 .8. But you might as well install update 6.14, which was the final update.
There is a free option for you too by downloading the latest version of adobe dng converter (Digital Negative (DNG), Adobe DNG Converter | Adobe Photoshop CC ) and converting all your sony raw files to dng. Those will work in Lightroom 4. Lightroom's conversion quality has improved by leaps and bounds between Lightroom 4 and the current Classic version (nominally version 7) so I would say it is worth it to put down the $10/month for Lightroom Classic in the photography plan but that is a personal decision.