When I go to move the Logic Pro sound library to an external drive, the drive is greyed out.

What kind of drive is it, how is it formated, how much free space is there?

1. Select the external drive icon in Finder and press Command-I to open its Info window.
2. Make sure the tabs "General", "More Info" & "Sharing & Permissions" are open, you can close all others tabs for now.
3. Press Command-Shift-4, your cursor arrow changes to a crosshair pointer with some small numbers (those are the crosshair coordinates on your display, but you can ignore that for now).
4. Place that cursor over the open Info window and press the Spacebar.
5. The cursor changes to a camera icon and highlights the Info window.
6. Click the Info window to make a screenshot of it.
7. On Catalina, you'll see a small preview of the screenshot in the bottom right corner. Click on it to open the preview, click "Done" to save it to your desktop.
8. Upload the screenshot to this thread.

The drive is either not formatted correctly or is too full. Verify format. Command-i as previously stated, and tell us what "Format" is set to. Very easy and quick. It must be set to "Mac OS Extended (Journaled)" or "APFS". Anything else is not usable. The drive would have to be reformatted (all data lost).

Good drive, plenty of space, it's not the drive, for sure. Even default permissions would allow this.

Now, what exactly do you mean the drive is greyed out? Where do you see this? Can you supply a screen shot?

^ Aha, there we have it.

How about the Ignore Ownership checkbox? Active or disabled? If disabled, that would explain everything.

Either you set it to "Ignore ownership", or you change the owner from System to your username.

Or, as Indyuser indirectly pointed out, if you use it as the Time Machine drive - then it'll also be greyed out.

Alright, that sounds good.

Now

What are the disk permissions? (see its Info window)

What else is on the drive already?

I'd suspect that there's already an older relocated sound library present, because that's a situation when Logic will gray out a drive.

Not necessary, I'll bet it's not permissions.

Are you using it as a Time Machine disk ?