Do I have to request permission from a website to use an image that they use to promote their product on their site?
It doesn't matter if the material is promotional or not. It doesn't matter if the material is freely available or not. What matters is copyright. You have to ask if you want to use copyrighted material.

On a different note, the company may really be unhappy if you use a screenshot which might be of bad quality (at least it's a pixel image) instead of a high quality, tailor made (possibly vector graphics) image for that purpose.