There will always be some people in any industry that just don't like tattoos to the point of potentially judging those who have them for the mere fact that they do have them. You can never avoid those people entirely, but they are in an increasing minority and not something that I would specifically worry about. They're certainly not a reason to not pursue a career as a solicitor.
In my experience attitudes to tattoos in the legal industry have followed attitudes generally to diversity and personal preferences, which is that there is more general acceptance. Even those people who would never get tattoos themselves (and I am one of them) usually have no issues with those who do have tattoos. I'm a barrister but can think of several examples in the last few years where I've had a solicitor with me on a trial who has had visible tattoos, both male and female. In those situations the fact of that solicitor having tattoos is something that is not so much a non issue as not worthy of note at all, and clearly their firm felt the same way.
There is, however, one very large caveat here, and that is that the law is a profession where image is important to the extent that clients will often not be satisfied with and will not want to instruct a lawyer that they do not think looks professional. And if a firm doesn't think your tattoos look professional and are concerned that clients might think the same, they won't want you to work for them. So it's not a prejudice against tattoos per se, but more a concern to the extent to which they can undermine a lawyer's professionalism.
Now, the general view as to what looks 'professional' has shifted over time. I'm very good friends with a male barrister who has long hair that often goes past his shoulders, which means he either has it like that in court or ties it back. A male barrister having hair like that would not have been tolerated at the Bar more recently than a lot of people think, both by colleagues and clients. But nowadays it doesn't matter, and has been no issue for him. At the same time, when it comes to all aspects of physical appearance, whether that be hair, tattoos, or something else, it is important that look professional. That depends to a degree on where your tattoos are and how visible they are, but also how large they are and what they are. Whilst I don't think there is an issue per se with you having tattoos, even visible tattoos, as a solicitor you do have to make sure that they look professional. A difficult thing to judge, I know, but it is something you have to have in mind.
So overall my broad advice would be that no, having tattoos in and of itself is not something that is likely to prevent you from becoming a solicitor. You may come across some who just flat out don't like it, but not to the extent that I'd actively worry about it. The main thing to have to bear in mind is that as a solicitor you do need be presentable and look professional. There can be a lot of personal choice in that, whether it's hair, piercings, jewellery and the clothes you're wearing. Tattoos are no different in that respect, but how others will perceive the tattoos you have is relevant. If your ambition is to get facial tattoos, for example, I think you're going to struggle. But providing you keep them broadly appropriate, I think you'll be fine.
All of that said, I am a barrister and not a solicitor. There may be others who have more direct experience of the attitudes towards tattoos in firms, and if you do get any replies you definitely need to be listening to those for obvious reasons.