Cant say whether any of these will work or not, but worth a try:
-Launching the game from either the LinksLauncher or LinksMMIII files (both found within the main Links 2003 folder)
-Compatibility settings: right click LinksLauncher and/or LinksMMIII and choose Properties > Compatility tab, and try various compatibility settings.
-Unplug any USB devices you might have connected, and see if the game works then. Especially certain models of sound devices are known to cause problems.
If neither of those work, and unless anyone else comes up with other suggestions, as a final measure I think it could be worth looking into this thread:
https://linkscorner.org/phpBB/viewtopic.php?f=9&t=788 However, I think that fix is traditionally more for issues that arise AFTER the game at least launches... Still worth a try if left with no other options.