The correct option is A become
The given sentence is conditional. It means that it presents a hypothetical situation that may or may not happen: "If you ___ a veterinary doctor", and its probable result "I will bring my dogs to you".
This type of sentence is the Type 1 conditional. (Type 1 conditional sentences present a probable result, hence it's in the simple future tense.)
In this, the if-clause is in the simple present tense and the main clause (probable result) is in the simple future tense.
So, the verb required to fill in the blank must be in the simple present tense.
Option A: 'Become' is a simple present tense verb. It refers to a possible condition: "If you become..."
Hence A is the correct option.
Option B: 'Were' is in the past tense whereas the if-clause is in the present. It does not match with the future tense verb 'will bring'. Hence B is incorrect.
Option C: 'Can be' presents a possibility or an ability. 'If you have the ability to become a doctor'. It does not agree with the context of the sentence. The sentence is a combination of condition and result, and not an ability or possibility. Hence C is incorrect.
Option D: 'Came to be' means 'developed into', 'came into existence'. It does not make the sentence sound complete. Moreover, this construction isn't used in conditional sentences. Hence D is incorrect.