Yes, it is possible. It is worth doing a google search along the lines of linkedin trainee solicitor then add the name of a big firm] and you will see people from all kinds of universities. Usually they have top grades however. Some firms require AAB as minimum A level grades and you need to pass at some first the Watson G test.
In years 2 and 3 of my law degree I volunteered weekly at the local law centred. I did moots. I got some regional local law firm work experience in my home town in the North , I applied to loads and loads of firms for a training contract in summer holiday after year 2 of my degree and from year 3 to about February of year 3 of my LLB when at last I got an offer. In other words it is a lot of work