How do I post images on Stack Exchange using Imgur? Do I have to make an account on Imgur?
You can use the built-in uploader without an Imgur account. You don't even need an account on Stack Exchange to use it.
You should see this icon in the formatting bar above where you type your post:
Clicking on it, or typing control + g, will open a dialogue where you can either paste an image/image URL (control + v), browse files on your system to upload, or type a URL to upload from.
On the app (at least for iOS), it's a camera icon and you have the option to upload from the camera roll take a photo and have that uploaded.
After the image is uploaded, it is automatically inserted into the post.
On some sites, users with <10 rep are not allowed to post images (see here). The image will still be uploaded to i.stack.imgur.com, but it will be displayed as a link, not an image. Links have the same format as images, except they don't have an exclamation point at the beginning.