Perhaps.
In the movie we see a toilet in the back ground, but in the short story it has that descripition of his releiving himself. It makes me think his trailer is much smaller, maybe even like a camper, or a caravan I think it is called in Europe in a English-centric way.
Like these?

The pic is from you

. I found it here:
When I think about the wind blowing down the curved length of the trailer, I have a vision of the type of trailers below. I imagine on a ranch they would have a few old trailers like these in leiu of a bunkhouse or something else.
BTW: this is the type of trailer I also picture when reading the words "the curved length of the traier"
On the topic of the sink peeing: I agree with most people that Ennis's trailer probably didn't have a toilet. This would be the practical answer.
Still leaves the question,
why Annie Proulx included it in her short story. I think for one it's part of the gritty realism which marks all her stories in Close Range. Rough-mannered and rough-spoken are the men she writes about as well as the style she chooses to do so.
Another reason is it shows Ennis's lack of material goods.
And it has something sad to it. A sense of not caring, of being impassive. Just letting the days go by ...
Which I think is Ennis's state of mind after Jack't death.