What error logs can I check to see why plugin updates are failing?

In wp-config.php, set the following (if not already set):

define( 'WP_DEBUG', true );
define( 'WP_DEBUG_LOG', true );
define( 'WP_DEBUG_DISPLAY', false );
@ini_set( 'display_errors', 0 );

This will make sure that on the site no error is shown, however, WordPress will log all errors in debug.log file inside the /wp-content/ directory.

NOTE: You must insert this BEFORE /* That's all, stop editing! Happy blogging. */ in the wp-config.php file.

You’ll find more details about it in this Codex Document.