1. Copy HTML code next to picture in Photobucket
2. Go to Livejournal entry page
3. Paste code
With Imgur, it goes like this:
1. Right click on picture in Imgur
2. Select 'Copy image address'
3. Go to Livejournal entry page
4. Click on 'Insert Photo'
5. Click on 'Paste URL'
6. Paste URL into box
7. Click 'Add To Post'
I don't know if I'm missing something, but that seems to be the only way to do it. To be honest, I'm not sure Imgur is actually meant for picture hosting in this way. It seems more of a depository for memes and reaction gifs or for looking at the albums on their own site.
I'm also not a fan of the layout of their site and the way you have to scroll down to see all the pictures one after another on one page rather than having them in a grid and a set amount per page. I don't know if that's something you can change in the settings, but if it is I haven't found out how to yet. And if you can't, it's going to make my folders with 100+ pictures extremely annoying to navigate.
So I'm not ready to commit and move all my pictures over quite yet. I'm a bit concerned by the fact that Imgur seems to be the most highly recommended site though. It makes me wonder if I'm going to be able to find anything better suited to my needs.