@ojwaffles Hello and welcome. The developers of Fatdog64 and the other well-informed local crowd haven't gotten to this yet, so I'll help as I can.
I don't craft webpages, so I'm not so familiar with the LAMP usage, but I looked for the packages and they are in the repository. You'll want to use the Gslapt package manager, Menu>Setup>Gslapt. Be sure to update it with the Update button, then enter your search term.
While you are waiting, be sure to read the help files (the blue question mark icon on the desktop). Fatdog64 is not a derivative of another distribution, but uses Linux From Scratch, and uses the Slackware package management tools.
As far as the GUI, what are you looking for? xfce is available as an SFS package, but Fatdog64 is a fairly minimal system overall, and doesn't bother with various bulky environments and such. There are various ways to adjust the look and feel of the existing GUI.
As the other posters said, in Fatdog64 you run as root, and don't need to use sudo to ask for permission to do things. And since it is not in the Debian family, it does not use apt. Also, unlike the Puppy family, bdrv is not a thing.
Your project will be uncommon, and you may learn more than you guessed working with Fatdog64.
Dan