Need a New Camera

Dear Adam,
If your camera supports (probably it does) the USB cable, then after taking pictures you may just connect the cable to the computer and access the pictures via cable and upload them to the internet. You will be eliminating SD card removal-insert copy paste process.
Or you have 3 options as below:
1) Take photos with a smartphone, upload them to the internet using the phone. You may also back up the photos and access them via your computer.
2) Take photos with a digital camera, transfer them to your computer and upload them to the internet. That's the thing you are doing right now. What you can do is just making the transfer method easier. Some cameras will support cable, bluetooth or wifi transfer. Photos taken by a camera are considered to be raw files so you have to edit them before publishing. That's why most cameras (even the best ones) won't support direct upload from a camera to the internet.
3) You may buy a digital camera with a WIFI support however you will still need a computer or smartphone to view the pictures and upload them to the internet. I know that there are few cameras that you can access the internet from however they are not that popular and are pricey I guess.
Hope that helps.