Can I move GarageBand to an external harddrive?

Yes, but if there are updates the updater will look for it in the Applications folder. To avoid such problems after you copy the application to the external drive replace the original with an alias to the real application on the external drive.

Select the real application. CTRL- or RIGHT-click and select Make Alias from the contextual menu. Copy the "Garageband alias" file to the Applications folder on your system volume. Rename it by removing the " alias" portion.

Or, select the real application, hold down the COMMAND and OPTION keys then drag the selection into the Applications folder. You should see a little curly arrow in the left bottom corner of the dragged icon.

Original....valuable.

You might also want to look into a Time Machine backup if you don't have something suitable in place already. You can keep the bulk there for access when you need them and just your working items on your HD.

As always tho, HDs are low cost these days, so it can be of great benefit to buy (much) more and expand for both the short and long term.

Good luck in any case.

Why?

As mentioned, it's preferable to leave it where it is, or use an alias. This is common practice with Apple supplied applications.