Would it be possible to download a GUI version of Emacs which includes its own terminal emulator for any terminal-related commands like executing shell scripts? I.e., it would not use the local system's terminal but would include its own sandboxed terminal.
Yes, there are many, and several of them come with Emacs. Which you will prefer depends on your precise needs.
eshell will both give you shells with no terminal–emulation at all.
shell runs your normal shell, while
eshell runs a shell made entirely in Emacs lisp, with better integration with the Emacs internals (for example, you can redirect the output of a command to an Emacs buffer, as well as to a file).
term will let you run a shell (or any other program) inside a terminal emulator. The one downside here is that most Emacs keybindings are altered while in this mode, so that you can operate ordinary programs that expect a wide variety of inputs.
If you are willing to install packages, then
vterm is a good one. The terminal emulation is compiled into a loadable module rather than being written in Emacs lisp, so it is faster.
You will want to read chapter 38 Running Shell Commands from Emacs in the Emacs manual for more information about
term and a few other related topics. You can also read the Emacs manual, and potentially many others, while inside of Emacs. Use
C-h i to open the Info viewer; it starts you out at an index of all available manuals.
In addition to the built-in option @db48x said, there are some third-party terminal emulator packages for Emacs.
If you like the idea of Shell, you can check out Coterm. It emulates a terminal emulator in Shell.
As @db48x said, Vterm is also an option if you have no problem installing C modules.
There is new option named Eat. It is pretty new to the list. In
M-x eat, it can also emulate terminal in Eshell. It has
complete mouse support, and shell integration like VSCodium (or
VSCode) is also available. [Disclaimer: I'm the author.]
All the options also works when Emacs is run in terminal.