Is there a website that gives you the average price of a hotel over a year?

It is not really possible to define a useful 'average', as the same rooms are offered for the same night for very different rates, depending on where you book, when you book, what options you chose (cancelable or not, etc.), and how you pay.
Some of these options are used more than others, and only the hotel knows which ones were used how often; they have however a strong interest not to publish this information, as it would result in people learning how to book the cheapest rates only.

So you can define many different ways of calculating your 'average', but for the majority of users it will be meaningless and/or misleading, as your assumptions would not apply to them.

Online rooms booking systems falls under the technique called "obfuscated price". So the price depends on many different factors. All of this is done to make an item not comparable to another. Thanks to that users cannot take advantage of some "price collector" which just simply give you cheapest price of that item.

And because of that you cannot have average price of a hotel over a year simply because there is no such thing as price.

The same "obfuscated price" technique is used in many other areas, e.g. flight prices, bank account fees, new automobile prices etc.