Item Drop Alert Disappearing and not coming back in TF2?

It's likely you dismissed the alert with an errant keystroke.

I tried going into the backpack to click on the alert but it didn't work either, and now I can't have my item.

Item alerts are just notifications, the item is already in your backpack when you receive the notification, so even if the alert stuffed up the item should still exist. Assuming you don't have any blank spots in-between items, the last item in your backpack will be the most recent addition.

Note that you need to dismiss the notifications in order to receive more drops, so if item drops have stopped entirely for you, then it might be worth raising a bug using the 'Report' option from the in-game main menu.

To check if you can still earn drops, you'll need to be 'actively playing':

In order for a player to receive an item drop, they must be "actively playing" i.e they meet all of the following criteria:

• Is connected to a VAC secured server.
• Is responding to in-game drop notifications while not utilizing external programs to do so.
• Is running only one instance of the game.
• Is not in textmode.

_{TF2 Wiki - Item Drop System}