Probably, is the answer. Actually the answer is really yes, of course we can. But would it be cost effective? That is the more difficult question.
Web sites are built in all sorts of different ways using a variety of technologies. Everything from the basic HTML site where each page is created separately and is generally 'static', i.e. doesn't change. To the highly complex data driven sites used by larger companies to manage multiple departments, data sources and who knows what else.
All these sites, assuming the were built in the last ten years or so, have a common factor, CSS. Cascading Style Sheets generally referred to as CSS for obvious reasons, are responsible for the layout and look of the website. This is where most of the work takes place to get the website behaving responsively or making it mobile friendly. In a style sheet we can set rules that adjust the design depending on the screen size the website is being viewed with. In principle straight forward but it can take quite a lot of time to work through to give the best result. There is no good instant fix so be very doubtful of anyone that suggests there is.
Which brings us to the question of cost. All websites need 'refreshing', like a new coat of paint every couple of years. This way keeping them looking good as expectations and design move on as does everything else. So wouldn't this be a good time to consider renewing your website using the latest technologies? We would create it so that in future the website could be redesigned or re-skinned, without having to redo all the content. The technology could be updated as the new stuff comes through. Worth a thought I would suggest.
Good websites don't come cheap so it is worth investing in one that will last you long into the future. Grow and adapt with you as your business expands and changes.