Making an AJAX call to refresh a table

The action in the Ajax call must be the name of the action on the add_action without the wp_ajax_nopriv.* for example:

var data = {
  action: 'fmsapi_refresh_hybrid_schedule',
  year: '2017',
  event: 'ONOSH',
  team: '4783'
};

You are targeting a class named select_team but this isn’t anywhere in your code. This is why your Js code cannot be fired on the change action.

Change this line:

// Output the form
echo '    <form>' .
'Select a Team:' .
'<select name="select_team" class="select_team">' .

You didn’t get a return value because you just return the a string and not echo out the value of the string.

in the function fmsapi_refresh_hybrid_schedule()

Replace

return frc_populate_hybrid_schedule($year, $event, $team);

With

echo frc_populate_hybrid_schedule($year, $event, $team);
exit;

Also there should be a nonce field + remove the space from the add_action and many more small issues that seems to be out of context now.