Typically, for veterinary medicine, only work experience achieved up to 3 years before the application (ie year 10 onwards if you intend on applying in year 13) will be counted towards the university requirements so I wouldn’t worry about getting work experience arranged before then. Once you’re at that stage keep contacting practices to try to get it arranged.
Some practices are a bit picky with the age of students and I found that a few places didn’t take anyone under the age of 16 (which was frustrating as I have a summer birthday). That’s not to say that nowhere will take you if you’re 14/15, there’s no harm in asking and the worst case scenario is that they’ll say no.