Technically a website is copyrighted. However, most people probably wouldn't mind you using it in the way you have described. To me, it would kind of be like taking the picture of a public storefront. That is of course assuming it is a public website. Even if they did mind, you can always claim a "fair use" defense, but that isn't an automatic or guaranteed defense.
Where it gets tricky is content within content. If the website contains a copyrighted photo then when you screenshot it, you are using the website AND the photo. You have to make sure you can do both.
Another potentially problematic situation is if you yourself are making money. Commercial considerations is one of the factors considered when evaluating a "fair use" claim.
You could always write the website and tell them what you would like to do and ask if it would be alright. If it is something which sends them traffic it is doubtful they will mind. That would be the best way to be safe, in my opinion.