Just tried a random website called VisaSponsors.co.uk who have a sponsor list (the government one doesn’t load properly).
While Hill Dickinson LLP is not there, a company called Hill Dickinson Business Services Ltd is (based at the same location too so likely the same firm in some way).
As this firm only has the skilled worker visa listed, I’d guess that the answer is no for TCs but yes once a certain PQE is reached.
I believe that firms have the option to sponsor VS and TC, however, they will not sponsor VS if they offer a direct TC route.
According to the FAQs from MC and US firms, they can provide the certification of sponsorship for TC. I assume that a trainee solicitor qualifies as a “solicitors’ clerk” and the salary meets the annual going rate/threshold of £30k, therefore, it would be easier to sponsor candidates in this case rather than for VS.
Found this thread 'Skilled worker visa threshold changes impact' that might helps:
https://www.thecorporatelawacademy....isa-threshold-changes-impact.8378/post-156789