No, it is not grammatically correct, however the meaning is clear and the mistakes are extremely minor. A much better way of wording it is: "I always miss my dogs when on vacation. So I brought my clay figurines with me, which look exactly like them." No need to say "on my vacation" here as it is implied. In the second sentence, you have mentioned bringing them with you twice, which is slightly redundant.

Yes, your sentence does have a grammatical sense but it would be better to say instead.... I miss my dogs now and then on trips during my vacations. So I carry my clay figurines with me which look just like my lovely dogs.

Grammatically speaking yes "it is better to use during than on