Is it possible to do a Postdoc without travelling? I am soon to finish my PhD and would love to do postdoctoral research (UK). However, for reasons I shan't go into, it is not viable for me to travel outside of my city and so I am unable to attend conferences/workshops/etc.

Is it feasible for me to obtain a postdoc position? Or do postdoc responsibilities necessarily entail travelling?
This is somewhat field dependent, but your academic career could be much more challenging if your disability or other circumstances prevent you from traveling. You may need to travel to job interviews at the very least. Not attending conferences to present your research will reduce the visibility of your work, and so you will struggle to get your ideas well known in your field. This could reduce your job prospects. You will need to have your research products well-advertised on the web in order to have any name recognition. Your research may also have to be more ground-breaking in order to garner recognition. 

If, for example, you work in Computer Science, conference publications are typically the most prestigious, so the inability to travel to present your work may mean that it is not published. 

I don't mean to be too discouraging, but I think you should reevaluate whether or not you can make 1 or 2 trips per year in order to present your work. It may do wonders for how hard your career is otherwise. 

On the plus side, you might also try talking to some conference organizers before you submit your work about presenting your results via video conference. I was at a workshop this week where a presenter couldn't make it due to a last-minute problem with his flight, and he gave a great presentation via Google Hangouts. He wasn't there to network, but at least his presentation happened. If your reasons for not traveling are due to disability (including mental disabilities like social anxiety, agoraphobia, etc.), the country where the conference happens may require that they try to accommodate your needs.