Is it appropriate to do this on my own, and pay out of my own pocket?
You would be using your own money to further company interests, not something you want to get into a habit of doing as that's a very dangerous slippery slope and you want to keep firm separation of your money vs company money.
If you want to show appreciation to your employees, get a budget to do it, and if you can't then thank you emails will have to do as clearly the company doesn't want to pay for the gift for all of the employees. And if just your employees get something, while rest of the company doesn't, well, that's not great optics for anyone, and you cannot really hide it as people talk, especially when they get stuff.
A bit of an update based on the other answers:
Yes, everyone want's free stuff and it's always nice, and everyone will love you for it. And yet it's not your job to distribute your own wealth to make people you manage happy, same as it's not your job to pay their salaries, health benefits and so on, and bonuses and premiums are as much part of that package as anything else.
So what you have to do is take the discussion up the stream to your boss/manager and see if there is a holiday budget, and if there isn't one - fight for one. That is well within your powers and responsibilities to do. And I know, that's an unpopular opinion - telling someone to not give employees free stuff, but then there is no more expectations of managers to spend their own money on their employers, than there is of anyone else. You get paid to work, not the other way around.