Is StarterGui an appropriate place to store LocalScripts?

ReplicatedStorage does not run scripts. It's called -Storage for a reason.

There are really only five places where LocalScripts can run:

1. A Player's Backpack, such as a child of a Tool
2. A Player's Character model
3. A Player's PlayerGui
4. A Player's PlayerScripts
5. The ReplicatedFirst service

It is obvious when you want to put a script in 1 and 5. But what about 2, 3, and 4? Here are some good rules of thumb:

For 2, put the LocalScript in the Character if a) You're manipulating the character itself and b) You expect the script to run again every time the player respawns.

For 3, put the LocalScript in PlayerGui if it manipulates the player's Gui.

For 4, put the LocalScript in PlayerScripts if you don't want it to reset whenever the player dies.

Because in your question, you're expecting the LocalScript to interact with the player's GUI, I would put it in the GUI.

tl;dr Yes, you can put it in StarterGui