When I launch the game the mouse cursor disappears. You can open doors in the lobby and whatnot, but it's pretty much blind luck since you can't see it.

Which of the X-Wing games has this issue?

1998 special edition

Use the game's launcher and enable "Use Software Cursor" in advanced options.

Where can I find the game's launcher? When I click advanced icons on the shortcut or the actual application file that's not an option.

It's in the game folder, called XWINGTIE.EXE.

You can also create a link and add the /softwarecursor option that way (that is what the launcher actually does as far as I can tell).
Or even use the latest xwahacker version to force it to be always enabled.