My sim has three awards and it's a shame to lose them by moving her to another lot.
When she won them, they were saved in her inventory and when she got home she brought them in her hand, but she couldn't save them anymore.
Is there a mod that allows you to save items that normally can't be saved in their inventory?

I don't know of a mod that does that. I do know of a hacking tutorial that teaches people how to make an object that stores the prize tickets from vacation, so that a family can pool their vacation tickets. It's possible that something similar could be made to store awards; I don't know. If you want to look into that, the tutorial is at http://www.woobsha.com/diy-HowDoI-VacationTokens.html

Or if you're just moving your sim because you want a bigger lot, you could try the lot size cheat. I've never used it, so I don't know if it actually works, but I've read that you can put "lot_size" and a number into the cheat console and that will resize your lot. Back up your game if you try it, of course!

Probably the easiest thing to do would be to hack the awards so that they become buyable, then re-buy the awards your sim had at her old house.

Edited to add: I just tried cloning one of the awards from Superstar and making it buyable, and it was super simple. You can do the whole thing in the Transmogrifier; you don't even have to edit anything with IFF Pencil. So making buyable versions of the awards your sim currently has is probably the easiest way to be able to move and still have your awards.

Thanks for your suggestions @Corylea <3<3 I decided to clone them, it was easy.
Although, when I went into the game, I found the award SimChoice looking like this:

It's a minor thing, it doesn't really bother me. But if there is a way to fix it I have no problem doing it

Look at thread about editing thumbnails. You can change the catalog picture using IFF Pencil 2, if it bothers you enough to go to the trouble of doing it.